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INTRODUCTION 



TO THE 



Differential Calculus 



BY 



/ 

EDGAR W. BASS, 

Professor of Mathematics in the U. S. Military Academy. 







WEST POINT, N. Y.: 

United States Military Academy Press, 

Private Robekt Ritcnius, Compositor. 

1887. 






Copyrighted, 1887. 
Edgar W. Bass. 



PREFACE 



Op his Pamphlet embraces the subjects and principles, which, in 
r$N^ the form of notes or lectures, have been given to the 
Classes of the United States Military Academy during the past 
several years, in order to assist them in understanding the 
Calculus. 

To the Officers of the U. S. Army, who have taught the 
subject with me, I am greatly indebted for many of the methods 
and demonstrations here presented. 

I am also under additional obligations to Lieuts. Edgerton, 
Gibson, and Alexander, for valuable assistance in preparing the 
sheets for the printer. 

Edgar W. Bass. 

West Point, N. Y., 
March, 1887. 



Greek Alphabet. 



A a Alpha 

B fi Beta 

r y Gamma 

A 8 Delta 

E e Epsilon 

Z C Zeta 

H j] Eta 

& $6 Theta 

I i Iota 

K u Kappa 

A A Lambda 

M m Mu 



iV y Nu 

S £ Xi 

o Omicron 

77 tt Pi 

P p Rho 

2" a 2, Sigma 

T r Tau 

T v Upsilon 

cp Phi 

X x Chi 

W $ Psi 

.Q a? Omega 



DIFFERENTIAL CALCULUS. 



CHAPTER I. 

CONSTANTS, VARIABLES, AND FUNCTIONS. 

1. In the Calculus quantities are divided into two general 
classes, constants and variables, 

A Constant is a quantity that has, or is supposed to have, a 
definite fixed value. 

A Variable is a quantity that is, or is supposed to be, continually 
changing in value. 

In general, constants are represented by the first letters of the 
alphabet, and variables by the last; but they should not, therefore, 
be confused with the known and unknown quantities of Algebra, 
which, in general, are constants. 

The same quantity may sometimes be either a variable or a 
constant, depending upon the circumstances under which it is con- 
sidered. Thus, in the equation of a curve, the coordinates of its 
points are variables; but in the equation of a tangent to the curve, 
the coordinates of the point of tangency are generally treated as 
constants. It is, therefore, necessary to determine from the circum- 
stances, or object in view, which quantities are to be regarded as 
variables, and which as constants, in each discussion. 

In general, any or all of the quantities represented by letters in 
any mathematical expression or equation may have definite values 
assigned to them, and be regarded as constants; or they may be 
considered as changing in value, and treated as variables. Thus, in 
the expression 47r; 2 , r is a constant if we suppose it to represent the 
radius of a particular sphere; but if r is considered as changing in 
value, it will be a variable. In the first case, 47ZT 2 is a constant, and 
measures the surface of a particular sphere; but when r is variable, 



6 CONSTANTS, VARIABLES, AND FUNCTIONS. 

47ZT 2 is also variable, and represents the surface of any sphere no 
matter how much it may increase or diminish. The formation of a 
soap bubble illustrates the latter case. 

It should not be understood, however, that we may in all cases 
treat quantities as constants or variables at pleasure without affect- 
ing the character of the magnitude represented by the expression or 
equation. For example, n is generally assumed to represent the 
ratio of the circumference of any circle to its rit&ius, which ratio is 
invariable. If a different value be assigned to 7t y the expression 
47rr 2 will not measure the surface of a sphere whose radius is r. 

In some cases variation in a quantity changes the dimensions of 
the magnitude represented by the expression or equation; in others 
it changes the position only; and again it may change the character 
of the magnitude. Thus, if we suppose R to vary in the equation 
(x— a) 2 -\-(y— J3) 2 =R 2 , we shall have a series of circles differing 
in size; but by changing a or ft and not R the position only will 
be affected. 

By changing b 2 within positive limits, the equation a 2 y 2 -{-b 2 x 2 =a 2 b 2 
represents different ellipses, but negative values for b 2 cause the 
equation to represent hyperbolas. In general, however, constants 
are supposed to have fixed values in the same expression, unless for 
a particular discussion it is otherwise stated. 

Functions. 

2. A quantity is a function of another quantity when its value 
depends upon that of the second quantity. Thus, ^ax is a function 
of 4, a, and x. In general, any mathematical expression which 
contains a quantity' is a function of that quantity. If, however, a 
quantity disappears from an expression by reduction or simplification 
the expression is not a function of that quantity. Thus, 
x 2 -\-(c-^-x) (c—x), —> and tan.* cot.*, are not functions of x. 

3. A function of a single variable is one whose value depends 
upon that of a single variable and varies with it. Thus, 

-^— 4 » s/^x 2 -f 2px> \og(a-\-x), sec*, 

in which x is the only variable, are functions of a single variable. 

Any function of a single variable is also a variable, and varies 
simultaneously with the variable. 



FUNCTIONS. 7 

4. The relation between a function of a variable and its variable 
is one of mutual dependence. Any change in the value of one causes 
a dependent variation in that of the other. Either may, therefore, 
be regarded as a function of the other; and they are called inverse 
functions. Thus, if x passes from the value 2 to 3, the function 2x 2 
will vary from 8 to 18; and conversely, x will increase from 2 to 3, 
if 2x 2 changes from 8 to 18. 

In the equation of a curve, the ordinate of any point may be con- 
sidered as a function of the abscissa, or the abscissa as a function 
of the ordinate. 

The function is considered as dependent, and the variable as 
independent; for which reason, the latter is called the independent 
variable, or simply the variable. 

Representing a function of x, as x 3 , by y, we have y=x s ; solving 
with respect to x, we have x=\/y; a form expressing directly x as 
a function of y. 

The difference in form in the following important examples of 
direct and inverse functions should be observed. 

Having, y=x n ; then x=\y. 

y=a+x; x=y — a. 

y 

y=ax: x=~- 

s ' a 

y=a*; x—\og & y. 

5. A state of a function corresponding to a value or expression 
for the variable is a result obtained by substituting the value or 
expression for the variable in the function. Thus, 

— 00 , — ida, — 2a, o, 2a, 16a, 00, 

are the states of the function 2ax? corresponding, respectively, to 

the values or expressions for x, 

— co , " — 2, — 1, o, 1, 2, 00, 

and 



\A 



2 



are the states of the function sin cp corresponding, respectively, to 
the expressions or values of cp, 



Tt Tt It 37T 

a' ~' o' ^J T' 27T. 

6 4 2 ' 2 



8 CONSTANTS, VARIABLES, AND FUNCTIONS. 

A function of a variable has an infinite number of states. It may 
have equal states corresponding to different values of the variable; 
and it may have two or more states corresponding to the same value 
of the variable. Thus, 

5 and i, 7±/^i2, 13 and 5, . 13 ±^24, 25 and 13, 

are. the states of the function 2x-\-i-±\ / A t x, Corresponding, respect- 
ively, to the values of x, 

1, 3, 4, 6, 9. 

Trigonometric functions have equal states for all angles differing by 
any entire multiple of 27t. 

In connection with any state of a function corresponding to any 
value of the variable, it is frequently necessary to consider another 
state of the function, which results from increasing the value of 
the variable corresponding to the first state by some convenient 
arbitrary amount. 

In order to distinguish between these two states of the function, 
the first is designated as a primitive state, and the other as its new or 
second state. 

Any arbitrary amount by which the variable is increased from 
any assumed value is called an increment of the variable. It is 
generally represented by the letter h, or k, or by A written before 
the variable; as, Ax, read "increment of '#". 

Let x f represent any particular value of x, and h, or Ax', its 
increment; then will 2ax' 2 and 2a{x f -\-1if, or 2a(x'-\- A x') 2 , repre- 
sent, respectively, the primitive and new states of the function 2ax 2 , 
corresponding to x' and its increment h, or Ax'. The general 
expression 2a(x-\-Jif represents the second state of any primitive 
state of the function 2ax 2 , and from it we obtain the second state 
corresponding to any particular primitive state by substituting the 
proper value of x. The increment of a variable is always assumed 
as positive. 

6. A function of two or more variables is one which depends 
upon two or more variables and varies with each. Thus, 

x%n\y, xy, x$, ylogx, x 2 + <yxy — $y } 
are functions of x and yj and 



x+y+z, v 2 +tan-' 2sin a (x s y), Vx^+y^+logz, 

are functions of x, y and z. Each variable is independent of the 



FUNCTIONS. 9 

others. Particular values or expressions may be assigned to one or 
more of the variables, and the result discussed as a function of the 
remaining variables. A function of two or more variables possesses 
all of its properties as a function of each variable. By substituting 
in the function 2X*-\-y, any, assumed value for y, as 5, the result 
2 * 2 +5 i s a function of a single variable. 

7. A quantity is a function of the sum of two variables when 
every operation indicated upon either variable includes the sum of 
the two. Thus, 

'}>c^/x±y, sin (x ±y), log (x±y), a x± y, 

and all algebraic expressions which may be written in the form 

A{x±yY+B{x±)) n ~ x + +H, 

in which A, B, etc., are constants, are functions of the sum of the 
two variables x and +y. 

Sax(x -\-y) n , -y 'x — y — iy, 's/x+y, x xJ r?, xsin(.r — y), 

are not functions of the sum of x and y. 

sin(* 2 ±j 2 ), A(x 2 ±y 2 Y, 3\og(x 2 ±y 2 ), %/ 2{x 2 ±jy 2 )+7a, 
are functions of the sum of the two variables x 2 and day 2 , but not of 
the sum of x and ±y. 

2(3;\/x+ay 2 ), cos 2 (b<\/x+ ay 2 ), 2 V \og(d^/x+ay 2 — y)i 
are functions of the sum of the two variables b^/ x and ay* . 

In any function of the sum of two variables, a single variable may 
be substituted for the sum, and the original function expressed as 
a function of the new variable. Thus, z may be substituted for 
(x-\-y) in the function a (x-y-yY, giving the function in the form 
az n . In a similar manner we may write 

tan(x— y)=tanz, a x +?=a z , 2a^/\og{x — y)=2a<y/logz \ 

but it must be remembered that z in the new form is a function of 
the two variables x and y. 

8. A state of a function of two or more variables, corresponding 
to a set of values or expressions for the variables, is the result 
obtained by substituting those values or expressions for the corre- 
sponding variables. Thus, 

— 20, — 6, o, 5, 25, 

are states of the function 4x-\-$y-\-2 corresponding, respectively, to 



IO CONSTANTS, VARIABLES, AND FUNCTIONS. 

the values or expressions for x and y, 

(-4—2), (—2,0), (—8, + 10), (o,i), (2,5); 

and 

o, V-, 1, a/3, <», 

are states of the function tan (x-\-y) corresponding, respectively, to 
the values or expressions for x and y, 

f 7C 7t\ ( n n\ (lit n\ / lt\ 

<*«»• U'i)' fee)' (r 9 -); (°-^ - 

Any function, in which all of the variables are independent, is. a 
variable, and has an infinite number of states. 

9. A function of several variables may be equal to some constant 
value or expression; in which case one of the variables is dependent 
upon the others. Thus, the first member of the equation 2x-\- $y= 7 
is a function of the two variables, x and jpy but x and y are mutually 
dependent. 

Any equation containing n variables expresses a dependence of 
each variable upon the others; and there are only n— 1 independent 
variables in such an equation. In other words, the number of 
independent variables in any equation is one less than the total number 
of variables. 

In any group of simultaneous equations, the number of independent 
variables is equal to the total number of variables less the number of 
independent equations. 

10. Functions are divided into two general classes, abstract 
and concrete. 

Abstract functions are subdivided into algebraic and transce?idental. 

11. An Algebraic function is one that can be expressed definitely 
by the ordinary operations of Algebra; that is, by addition, sub- 
traction, multiplication, division, formation of powers with constant 
exponents, and extraction of roots with constant indices. 

12. Certain algebraic functions have particular names based 
upon peculiarities of form. 

A rational function of a variable is one in which the variable is 
not affected by a fractional exponent. 



EXPLICIT AND IMPLICIT FUNCTIONS. II 

An integral function of a variable is one in which the variable 
does not enter the denominator of a fraction, or in other words, is 
not affected by a negative exponent. 

x m +Ax m ~ 1 +Bx m - 2 + Gx+H, 

in which m is a positive integer, and A, B, etc., do not contain x, 
is a rational and integral function of x. The coefficients A, B, etc., 
may be irrational or fractional. 

A rational integral function of a variable is also called an entire 
function of that variable. 

' A linear function of two or more variables is one in which each 
term is of the first degree with respect to the variables. 

Thus, 2x-\-i,y-\-'iz is a linear function of x, y and z. 

A function is homogeneous with respect to its variables when 
each term is of the same degree with respect to them. 

A linear function is a homogeneous function of the first degree. 

13. Other divisions of functions, based upon form or properties, 
are of frequent use. 

Explicit and Implicit Functions. When a function is expressed 
directly in terms of its variable or variables, it is an explicit func- 
tion; otherwise it is an implicit function. 

Thus, in the equations 

}'=2x' 2 +3z, y=tan 2 x, v=3 x , y=log2ax z , y—f{x, z), 
y is an explicit function of the variables in the second members, and 
in the equations 

a' i y 2 -\-b-x~=a 2 b'', y t =\ogx 2 , y z =r 2 — x'" , <\/y=cotx, y n =f(x), f(y,x)=o, 
y is an implicit function of x. 

The relation between an implicit function and its variables is 
sometimes expressed by two equations. Thus, y=su, u 2 = ^/x f 
in which y is an implicit function of x. 

y=f( u ), it — cp{x)\ and y=f(u), x=cp(u), 
are forms expressing y as an implicit function of x. 

14. Increasing and Decreasing Functions. A function that 
increases when a variable increases, and decreases when that variable 
decreases, is an increasing function of that variable. Thus, 2x, jx 3 , 
2 X , —-> are increasing functions of x. 



12 CONSTANTS, VARIABLES, AND FUNCTIONS. 

A function that decreases when a variable increases, and increases 
when that variable decreases, is a decreasing function of that variable. 
Thus, -' (c— x) 3 , — -, are decreasing functions of x. 

x v ' ax z & 

Functions are sometimes increasing for certain values of the 

variable, and decreasing for others. Thus, (c— x) 2 is an increasing 

function for all values of x greater than c; but decreasing for all 

values of x less than c. 2ax is an increasing function when x is 

positive, and decreasing when x is negative. The positive value of 

y= + Vr 2 —x 2 , is an increasing function for values of x from — r to o, 

but decreasing for values of x from o to -fr. The negative value of y 

is a decreasing function for negative values of x, and increasing for 

positive values of x. 

15. Continuous and Discontinuous Functions. A function is 
continuous between states corresponding to any two values of a 
variable, when it has a real state for every intermediate value of the 
variable; and as the difference between any two intermediate values 
of the variable approaches zero, the difference between the corre- 
sponding states approaches the same limit. Otherwise a function is 
discontinuous between the states considered. 

A continuous function in passing from any assumed state to 
another must pass through all states intermediate to those assumed; 
but it may 'have intermediate states greater or less than the states 
assumed. Thus, the function Vr 2 — x 2 is continuous between the 
states o and -V3, which correspond to x= — r and x= - ; but it 
is greater when x=o than either of the states considered. 

An imaginary or infinite state, or the omission of any state be- 
tween the extreme states considered, interrupts the continuity. 

A function always continuous changes its sign only by passing 
through zero; but a discontinuous function may change its sign 
without passing through zero. 

Entire functions of a variable are always continuous. 

^^Jipx is continuous between states corresponding to x= — o and x=co . 

=t-<y/tf 2 — x* is continuous between states corresponding to x= — a and x=+a. 

b j x= — 00 and x— — a. 

i~V x' a — a 2 is continuous between states corresponding to i x=a an( j x —^ 

but is discontinuous between states corresponding to x= — a and x=a. 



TRANSCENDENTAL FUNCTIONS. 



13 



16. A Transcendental function is one that cannot be expressed 
definitely by the ordinary operations of Algebra. 

In general, a transcendental function maybe expressed algebraic- 
ally by an infinite series. 

Transcendental functions are of four kinds, exponential, loga- 
rithm ic, trigonometric, and inverse trigonometric. 

An Exponential function is one with a variable exponent; as, 

a x , f-4-i) ' <? x — 2 ex. 

A Logarithmic function is one that contains a logarithm of a 
variable; as, 



log:. 



log (a+J'), 



2ax~- 



logx 



A Trigonometric function is one that involves the sin, or cos, 
or tan, or cosec, etc., of a variable angle; as, 

x — sin.r tanx — x 



cot x, 



x — sin x 



An Inverse Trigonometric function is one that contains an angle 
regarded as a function of a variable sin, or cos, or tan, etc. 

Sin-M', tan- 1 )', cosec- 1 /, read "the angle whose sin is y"j "whose 
tan is y"j "whose cosec is y"j are symbols used to denote such 
functions. Having given y= versing, then x=\ersm- 1 y; and if 
u=cosy, then y=cos- 1 u, etc. 

1 7. It should be observed that although the number of different 
abstract functions of a single variable is infinite, they involve but 
ten elementary or simple forms, five of which are the inverse of 
the others. 

Representing the direct functions by y, and the variable by x, 
the forms are as follows: 

Algebraic. 



Direct. 




Inverse. 




y=x+a. 


Sum. 


x=j—a. 


Difference. 


y=ax. 


Product. 


y. 

x= 

X 


Quotient. 


y=x n . 


Power. 

Transcendental. 


•r=Vj. 


Root. 


y=a*. 


Exponential. 


x=\og a y. 


Logarithmic. 


j'=sinx. 


Trigonometric. 


x—- sin- 1 y. 


Inverse Trigonometric 



14 CONSTANTS, VARIABLES, AND FUNCTIONS. 

18. Functional Notation. A function of any quantity, as x, 
is represented thus, f(x), read " function of x". Other forms are 
also used, as, f(x), F(x), F 1 (x), <p(x), <p L (x) y tp(x), i/,\(x). 
The quantity is written within the brackets, and a letter, as /, or F, 
or cp, etc., is placed before the brackets to represent the operations 
involved in any particular function. 

Having assumed the exterior letter, its significance remains un- 
changed throughout the same subject. Thus, if -^— is represented 
by f(x), f indicates that the quantity within the brackets is multi- 
plied by a, and that the product is divided by i plus the quantity. 
Hence, 

ay az am la a sin <v 

/W=7TT /«=7^r /«=I+^' /W=1+T / (sin< P)=TTii^ - 

Like functions of different quantities, when considered in the 
same subject, require the same exterior letter. In order to represent 
different functions of the same quantity, the exterior letter is changed, 
but not the letter within the brackets. Thus, if F(x) is selected to 
represent 2 \'bx; then some other form, as F 1 (x), or cp(x), etc., 
should be taken to denote 4<rx z -\-2x. 

Different functions of different quantities are represented by 
forms which have different letters within and without the brackets. 
Thus, \/x 2 —a 2 , and -^-^ may be denoted by f(x), and ip(y), 
respectively. 

A function of x 2 is written f(x 2 ), or F(x 2 ), etc., and the square 
of a function of x is designated by/(^) , or cp(x) , etc. 

$c\/my 2 may be expressed as a function of my 2 by some form, 
as /(my 2 ), or /' (my 2 ), etc. 

Having represented az 2 by f(z), and $c \/az 2 by F(az 2 ), we may 
write y i/az~ 2 =F(az 2 ) = F[f(z)']. 

. Similarly, having a x =<p(x), and b \ / a K —ip(a x ), we write 

U ^-yj^/g* =/ , A [ ^J \ > . n whkh lj) \ jp{pc) '] =b ^ m 
idb^/a* V J 

An expression containing several different functions of a variable, 
as 2ax 2 — log #+3 sin x, may be considered as a function of the 
several functions of the variable, and represented thus, 
F\_f(x), <p(x), f{x)]. 



FUNCTIONAL NOTATION. 15 

Different functions of the same variable, as x, are frequently 
denoted by the symbols X, X\ X\ etc.; in which case a function of 
the several functions may be indicated by f(X, X', X", etc.). 

(p[F(y), ip(x), f(z)~\ represents a function of three different 
functions of different variables. 

Representing different functions of x by X, X', X", and various 
functions of y by Y, Y', Y", a function of the several functions of 
x and j would be indicated by F(X, X', X\ Y, Y', Y"). 

Functions of two variables are denoted thus, f(x,)')i f { x O')> 
F(y,z), cp(x,y), ip(x,z), if\(x,z), etc.; and functions of three 
variables by F(x,y, z), ip (r, s, /,), etc. 

Functions of any number of variables are indicated similarly by 
writing all the variables within the brackets or parenthesis. 

In all cases the like exterior symbols have the same significance 
in any one subject. 

Thus, if f (x,y)=ax-\-fy; then f(s,t)=as-{-bt; f(2, 3)= z 2a-\-7,dy 
f(o,7ii)=b??i. 

Having cp (x,y, z) = 2x — cz-\-y 2 ; then cp (r, s, t) = 2r— cs-\-t 2 . 

Functions are frequently represented by single letters; thus 
± \/R~— x' 2 may be represented by y, giving y=± ty'R 2 —x 2 ; 
and/(a-,_r) by z, giving z=/(x t y). 

Illustrations. 

1. Having f(x)=x m +I > x m - 1 + Qx m - 2 + + U, in which/', Q, etc., do not 

contain x; then, 

/(5) = 5 m +^5 m_1 + <25 m_1 + - • -- + U. 
/(3^)=( 3 ^) m -f-^(3^) m - 1 + ...- + U. 

f{a—x)={a—x) m +P{a—x) m - 1 + + U. 

f(o)=o m +Po m - 1 + +U. 

/(^)=(^)»+ J P(^ 8 ) m " 1 + -- • •+#"■ 

Having then, 

2. cp (a)=4a*+ca; cp(x+y)=4(x+y)°~ +c(x+y), 

3. 2b(ax 3 )=4(ax s ) 2 +c(ax*j; if>(smB)=4sm 2 B+csinQ. 

4. F{x)=a*; F(x+v)=a*+r=axXay=F{x)XF(y). 



z- 



F{xy)=a*r; F{x)' =F(y) =(«*)' =( fl y) =F(xy). 



l6 CONSTANTS, VARIABLES, AND FUNCTIONS. 

6. If <p(z)=\/zy tp(x) = $ax; and F(w)= C V 7t/ — M* ; 



then K0 !>(*)];= n/ - 

1 5a V 2— 2/ 



37^—2/ 



Having then, 

7- /(Xfld—axT-tyj f{)', x )~ a y — bx; and f(z,z)=az — bz. 

8. ^(j, z)-=a.y+^/z; ip(a,b)=a a + yd; and ^(xjj^aHyj' 

9. cp(x,y, z)=?,x — logy+tanz; <p(r,s, t) = y — log^+tan t. 

10. F(x,y, z)=x 3 +2y+\/z J - F(2, — 8, 64)=o; and ^(o, o, i6)=4. 



11. 



r -1 3 a 

iP [/(a), ^(6), ?>O0]; in which, /(*)=—; 



tan 6 2y n 

12. /(*, v, 2)=»j M + b ?+ M =»r w X»« bJ X»« M =/(^ o, o)X/(o,j, o)X/(o, o, 2). 

13. If <pCr,_y) = 2;c+sinjj// and ^{z)—3\/z; then ^|j2> (x^J^Vs x+sinjj'. 

14. If /"(•*"» jj/, z) = "jax 2 yz; and ^(j / )=V JJ/ S / and (p(x)=a x y and ip(z) = 2s; 
then ^ | 



L(^[/(x, 7> 2)]) 



2a v 



19. Lines. Any portion of any line may be considered as 
generated by the continuous motion of a point. 

Let s represent the length of a varying portion of any line in the 
coordinate plane XV, of which the equation in x and y is given. 
s depends upon the coordinates of its variable extremities, and 
varies with each; but the equation of the line establishes a depend- 
ence between these coordinates. Hence, s is a function of one 
independent variable only. 

If the line is in space, its two equations establish a dependence 
between the three coordinates of its extremities, so that one only is 
independent. 

The same result will follow if a system of polar coordinates 
is used. 



GEOMETRIC FUNCTIONS. 



17 



20. Geometric Representative of a Function of a Single Variable. 

By laying off upon the axis of abscissas assumed values of any 
variable, and upon the corresponding ordinates, distances represent- 
ing the corresponding states of any given function of that variable, 
a line may be determined, the coordinates of whose points will have 
the same relations as those existing between the corresponding 
states of the function and values of the variable. 

Hence, every function of a single variable may be geometrically 
represented by the variable ordinate of a line, of which the variable 
abscissa represents the variable. 

It follows, that the relation between a function and its variable 
may be expressed analytically by the equation formed by placing 
the function equal to a symbol representing the varying ordinate. 
Thus, placing the function 7.x 2 +3 equal to y, we have y = jx' 2 -\-$, 
which expresses the relations between the variable coordinates y 
and x, and therefore between the function 7^ 2 -f 3, and its variable x. 

The equation thus obtained is that of a line whose ordinate, not 
the line, represents geometrically the given function. 

A function of a single variable, which is of the first degree with 
respect to the variable, will be represented geometrically by the 
ordinate of a right line. 



The ordinates PM, and PM' of the curve 
MQM'\ represent geometrically two different states 
of the function corresponding to the same value of 



the variable, 



§ 5- 





The ordinates PM, PN, and SO of 
the curve MNO, represent geometric- 
ally equal states of the function cor- 
responding to different values of the 
variable, § 5. 



It is important to notice that the function represented by a line 
is not, in general, the function represented by its ordinate. The 
problem of determining a line which represents a given function 
of a single variable ; or a function which is represented by a 
given line, is not, in general, a simple one. Therefore, the method 



1 8 CONSTANTS, VARIABLES, AND FUNCTIONS. 

of representing geometrically a function of a single variable by the 
ordinate of a line is generally adopted. 



21. Surfaces. Any portion of any surface maybe considered 
as generated by the continuous motion of a .line. 

M Let u represent the area of a varying portion 

of the surface generated by the continuous 
. motion of the ordinate of any given line in the 
plane XY. 

-X u depends upon the coordinates of the 
variable extremities of that portion of the given line which limits it, 
and varies with each; but the equation of the given line establishes 
a dependence between these coordinates. Hence, u is a function 
of but one independent variable. 





22. Let r=f{v) be the polar equa- 
tion of any plane curve, as DM, referred 
. to the pole P, and the right line PS. Let 
r u represent the area of a varying portion 
of the surface, generated by the radius 
vector revolving about the pole, u will 
change with v and r; but v and r are 
mutually dependent. Hence, u is a func- 
tion of but one independent variable. 




23. Let any line in the plane XY, as A M, 
revolve about the axis of X. It will generate a 
surface of revolution. 

The same surface may be generated by 
~* the circumference of a circle, whose centre 
moves along the axis X, with its plane perpendicular to it ; 
and whose radius changes with the abscissa of the circle, so as to 
always equal the corresponding ordinate of the curve AM. The 
radius of the generating circumference is, therefore, a function of 
the abscissa of its centre. Hence, the generating circumference, 
and any varying portion of the surface generated, is a function of 
but one independent variable. 



GEOMETRIC FUNCTIONS. 



*9 



24. The area of any surface with two independent variable 
dimensions is a function of two independent variables. For exam- 
ple, the area of any rectangle with variable sides, parallel respectively 
to the coordinate axes X and Y, is a function of the two independ- 
ent variables x and y. 

25. Having any 
surface, as A TL y let 
ABCD—u be a portion 
included between the 
coordinate planes XZ, 
YZ, and the planes 
DQRand ^/^parallel 
to them respectively. 
Let OI>=x, and OQ=y 
be independent varia- 
bles, u will depend 
upon x,y and zj but the 
equation of the surface 
makes z dependent up- 
on x and y. Hence, u 
is a function of but two 
independent variables. Similarly, it may be shown that any varying 
portion of the surface included between any four planes, parallel 
two and two, to the coordinate planes XZ and FZ, is a function 
of but two independc7it variables. 

26. Geometric Representative of a Function of Two Variables. 

By laying off upon the axes of x and y, respectively, assumed values 
of any two variables; and upon the corresponding ordinates, distances 
representing the corresponding states of any given function of the 
two variables, a surface may be determined having the same relations 
between the coordinates of its points, as those existing between the 
corresponding states of the function and values of the variables. 

Hence, every function of two variables may be geonietrically repre- 
sented by the variable ordinate of a surface, of which the variable 
abscissas represent the variables. 

It follows, that the relations between such a function and its 
variables may be expressed analytically by the equation formed by 




20 CONSTANTS, VARIABLES, AND FUNCTIONS. 

placing the function equal to a letter representing the varying 
ordinate. Thus, placing the function 2x 2 -\-^y-{-j equal to z, we 
have z=z2x 2 -\-$y-\-']; which expresses the relations between the 
variable coordinates z, x and y, and therefore between the function 
2x2 -\~5y-\-7, and its variables x and y. The equation thus obtained 
is that of a surface whose ordinate represents the given function 
geometrically. 

It is important to notice that the function represented by a sur- 
face is not, in general, the function represented by the ordinate of 
the surface. 

The problem of determining a surface which represents a given 
function of two variables; or a function which is represented by a 
given surface, is not, in general, a simple one. Therefore, the method 
of representing geometrically a function of two variables by the 
variable ordinate of a surface is generally adopted. 

27. Volumes. Any portion of any volume may be considered 
as generated by the continuous motion of a surface. The form of the 
surface, and the law of its motion determine the nature and class 
of the volume. 

Let any plane surface included between 
any line in the plane X Y, as A M, and the 
axis of X be revolved about X. It will gener- 
ate a volume of revolution. The same volume 
may be generated by the circle, whose centre 
moves along the axis X, with its plane perpen- 
dicular to it; and whose radius changes with the abscissa of the 
circle, so as to always equal the corresponding ordinate of the curve 
AM. The radius of the generating circle is, therefore, a function 
of the abscissa of its centre. Hence, the generating circle, and any 
varying portion of the volume generated, is a function of but one 
independent variable. 




GEOMETRIC FUNCTIONS. 



28. Having any 
volume, as A TZ, 
bounded by a surface 
whose equation is 
given, and the coordi- 
nate planes, let 
ABCD-OJV=V, be 
a portion included be- 
tween the coordinate 
planes XZ, Y Z, and 
the planes DQR and 
BPS, parallel to them 
respectively. 

Let OP=x, and 
OQ—y, be independent 
variables. V will 

depend upon x, y and z; but the equation of the surface makes 
z dependent upon x and y. Hence, V is a function of but two 
independent variables. 

In a similar manner, it may be shown that any varying portion of 
the volume included between any four planes, parallel two and two, 
to the coordinate planes X Z and YZ, is a function of but two 
independent variables. 




29. Any volume with three independent variable dimensions is 
a function of three independent variables. For example, the volume 
of any parallelopipedon with variable edges parallel, respectively, to 
the coordinate axes, X, Y and Z, is a function of x, y and z; all of 
which are independent. 



CHAPTER II. 

PRINCIPLES OF LIMITS. 

30. The Limit of a variable* is a fixed quantity or expression 
which the variable, in accordance with a law of change, continually 
approaches but never equals; and from which it may be made to 
differ by a quantity less numerically than any assumed quantity 
however small. 

As an example, take the variable expression , and increase 

x continually. Under this law, will continually approach, but 

never equal, unity, x may be taken so large that the difference 

between the corresponding value of and unity, will be less 

numerically than any assumed number however small. Unity is 

therefore the limit of the variable — ^— , under the law that x in- 

creases and becomes greater than any assumed number. This may 
be indicated as follows, 

limit T i 



&£]«. 



xm^ao Li+x J ' i + oo ' 

or, having represented by f(x), we may write /(co) = i. 

y\ is the limit of the repeating decimal fraction 0.272727 , 

under the law that the number of places of figures is indefinitely 
increased. 

The circumference of a circle is the limit of the perimeter of an 
inscribed regular polygon as the number of its sides is continually 
increased. The radius is the limit of the apothem, and the circle, 
of the polygon, under the same law. 



* In this chapter the term variable is used in its general sense § 1, and includes 
all functions of variables. 



PRINCIPLES OF LIMITS. 23 

In all cases, when referring to the limit of a variable, it is necessary 
to give the law; for the limit depends not only upon the variable, 
but also upon the law by which it changes. Under a law, a variable 
has but one limit; but it may have different limits under different 
laws. 

The manner in which a variable approaches a limit depends upon 
the law. It may be less than the limit, and continually increase; 
or it may be greater than the limit, and continually decrease. A 
variable sometimes approaches a limit by values alternately greater 
and less than the limit. 

Examples of the latter class may be found in the n successive 
approximating fractions of a continued fraction. The continued 
fraction being the limit of the successive approximating fractions 
under the law that n increases. 

is the limit of any number, which under a law of change, 
becomes less numerically than any assumed number however small. 

cc is the limit of any number, which under a law of change, 
becomes greater numerically than any assumed number however 
great. 

and 00 are neither numbers nor measures of quantities. 

Limits, as denned, include all results obtained by the substitution 
of or oc for any variable quantity or quantities which enter any 
expressions. Thus, A is the limit of A-\-h as h approaches 0. 

Since infinity is indefinite, two infinities cannot, in general, be 
compared with each other. 

Expressions, such as 

a cc .. . 

i-f-co J ' 

are symbolic forms indicating the limits of certain variables, and the 
law of change. 

The statement that one number or value is infinitely great as 
compared with another, is inaccurate. A number, however small, 
cannot be neglected or omitted in comparison with any other, how- 
ever great, without error. In applied mathematics numbers or values 
are sometimes neglected in comparison with others when approxi- 
mate results are sufficiently accurate for the object in view. 

Any variable, which under a law approaches zero as a limit, is 
called an infinitesimal. 



24 LIMITS. 

As any variable approaches its limit, under a law, the difference 
between it and the limit approaches zero as a limit. Hence, the 
difference between any variable and its limit is an infinitesimal under 
the same law. 

Let u be any variable, and C a constant which is its limit under 
a law. Let € represent the infinitesimal C — u; then, 

€=C — u . . . (i), or u = C — e . . . (2), or C=u+£. . . . (3), 
in which the sign of e depends upon C and u. 

That is, 

i°. A constant is the limit of a variable when the difference 
between the constant and the variable is an infinitesimal under the 
law. 

2 . A variable is always equal to its limit under a law, minus the 
infinitesimal which is the variable remainder obtained by subtracting 
the variable from its limit. 

3 . A constant is the limit of a variable when it is the sum of the 
variable and an infinitesimal under the law. 

An infinitesimal is not necessarily a small quantity in any sense. 
Its essence lies in its power of decreasing numerically, in other words, 
in having zero as a limit; and not in any small value that it may 
have. It is frequently denned as " an infinitely small quantity "j 
that is not, however, its significance as here used. 

In representing infinitesimals by geometric figures they should 
be drawn conveniently large; and it is useless to strain the imagi- 
nation in vain efforts to conceive of the appearance of the figure 
when the infinitesimals decrease beyond our perceptive faculties. 
Usually one or two auxiliary figures representing the magnitudes at 
one or two of their states under the law, give all the assistance that 
can be derived from figures. 

3 1 . Theorem I. A variable with a constant sign cannot have a 
limit with a contrary sign. 

For suppose f(x) is always positive, and that limit /(#) = — C. 
From the definition of a limit, § 30, f(x) may be made to differ from 
— C by a value numerically less then C. It would therefore become 
negative, which is contrary to the hypothesis. In a similar manner, 
it may be shown that a variable always negative cannot have a 
positive limit. 



PRINCIPLES OF LIMITS. 25 

Theorem II. If the corresponding values of any two variables 
approaching their limits under a law, are always equal, the variables 
have the same limit. 

Let u and v represent any two variables giving always, under a 
law, u = v. 

Suppose C to be the limit of u; then u = C — e, in which € is 
an infinitesimal under the law. 

Substituting in above we have 

C— e=v, or C=v+e. 
Hence, C is the limit of v under the same law.* 

Theorem III. If the difference between the corresponding values of 
any two variables, approaching their limits under a law, is an infinitesi- 
mal, the variables have the same limit. 

Let u and v represent any two variables giving 
u — v = 6 x , or u = v-|-6 v , 
in which S is an infinitesimal. 

Let C be the limit of u, then u = C — e, in which e is an in- 
finitesimal. 

Substituting in above we have 

C— e=v+d, or C— v=6+e, 
the second member of which is an infinitesimal. Hence, C is the 
limit of v. 

Theorem IV. The limit of the sum or difference of any number of 
variables is the sum or difference of their limits. 

Let u, v, w, etc., represent any variables, and A, B, C, etc., their 
respective limits; then 

u = A — e, v = B — d, w=C — go, etc., 
in which e, <5, go, etc., are infinitesimals. 

Adding, or subtracting, the corresponding members we have 

±u±v±w±&c.= ±A±B±C±&c.T£TSTgjT&c. 
Hence, Theorem II, 

limit [±u± V±w± &c.]= ± A ± B ± C ± &c. 
= ± limit u± limit v± limit w±&c. 

* Hereafter, in order to avoid the frequent repetition of the expression ' ' under 
the law", it will be assumed, unless otherwise stated, that the changes in all the 
variables considered together, or in the same theorem, are due to one and the same 
law; and that all variables and their functions are continuous between all states 
considered. 



26 LIMITS. 

Theorem V. The limit of the product of any two variables with 
finite limits, is the product of their limits. 

Let u and v represent any two variables having the finite limits 
A and B respectively; then 

u = A — €, and v = B — 6, 
in which 8 and S are infinitesimals. 

Multiplying, member by member, we have 
uv=AB— Be— Ad+ed. 
Hence, Theorem II, 

limit [uvj= A B = limit u. limit v. 

Cor. The limit of any power or root of any variable with a finite 
limit, is the corresponding power or root of its limit. 
Thus, 

limit u m = (limit u) m , and limit u>» = (limit u)*». 

Theorem VI. The limit of the quotient of any two variables with 
finite limits, is, in general, the quotie?it of their limits. 

With the same notation as above we obtain by division, 

u A — e A Ad — Be 

+ 



v ~ B— 8 ~ B ' B (B— d) 
Hence, if limit v = B is not zero, we have 

limit u 



limit 



LvJ B 



limit v 
If both u and v are infinitesimals, the theorem fails; as it should 

since limit - can have but one value, 8 30; whereas ,! m ! u = -> 
LvJ ' * ° ' limit v 

may have an infinite number of values. 

Hence, we cannot write limit - ==r?^ — = -• 

LvJ limit v 

This failing case of the theorem is particularly important, as it 

explains a subsequent result upon which the main application of 

the principles of limits to the Calculus is based. Some examples are 

given to illustrate it. 



limit 



limit 



limit 



r (Vi+*— 1) (Vi+h-i) ~i 

t [" i+e — 1 ] limit T 1 I _ T 



efe t o[y / i+^— 1] 
whereas umit r 1 — a- 



PRINCIPLES OF LIMITS. 



27 



2°. Through the point A, with- 
out the angle M N N' , draw right 
lines APS intersecting the sides 
AfN and N' A T nearer and nearer 
to N. The segments P JV and 
SA T are infinitesimals under the 

, , , limit PA 

law, and we have 



limit SN 








The ratio 
sin ASP 



PA 



— is always equal to the corresponding value of 



sin XPS 

Hence, limit — 1 
determinate. 



,. .«. Tsn AS PI 
limit - — — — - 
Lsm APS J 



sin BAA 
sin; CNQ 



which 



Y 




M' 






u 


^ 


^G^^ 


Q' 

V' 

P' 




y 




y 
p 


AX 


X 



The following example not only illustrates the case under consideration, but it 
also establishes a principle of great importance. 

3 . Represent any func- 
tion of any single variable, 
as x, by j<, giving y=f(x). 

Let BMW be the 
curve whose ordinate repre- 
sents the given function, 
§ 20. Take any state of the 
function, as P M corre- 
sponding to x=OP, and 

increase x by PP' represented by Ax. Draw the ordinate P' M' 
and the secant MM'. Through M draw MQ' parallel to X. Q'M', 
denoted by Ay, will represent the increment of the function corre- 
sponding to Ax. 

%—=r: = — = tan Q' MM J will be the ratio of the increment of 

PP' Ax ^ 

the function to the corresponding increment of the variable. 

At M draw M T tangent to the curve. Then, under the law that 
Ax approaches zero, the secant MM' will approach coincidence 
with the tangent M T, and the angle Q'MM' will approach the 
angle Q'MT, or its equal X H T, as a limit. 

Hence, 



limit 



oLaxJ 



limit 

Axm-^0 



[tan Q'MM '] = tan XHT. 



2 8 LIMITS. 

That is, the limit of the ratio of any increment of any f miction of 
a single variable to the corresponding increment of the variable, under 
the law that the increment of the variable approaches zero, is equal to 
the tangent of the angle made with the axis of abscissas by a tangent, to 
the line whose ordinate represents the function, at the point corresponding 
to the state considered. 

When M f coincides with M the secant may have any one of an 
infinite number of positions other than that of the tangent line M T, 
for the only condition then imposed is that it shall pass through M. 
Therefore, while llHllt — is definite, and equal to the tangent of 

Z\ JCW} — ^U L Z\ OC _J 

the angle that the tangent line at M makes with X, 7?—? — - = - 

to & ' limit A.r 

indicates that the tangent of the angle which the secant makes with 
X becomes indeterminate when M' coincides with M. 

Limit \^- is, therefore, one of the many values that ^! — - 
LaxJ ' J limit Ax 

may have under the law. 

It should be observed that if limit -J =1, then limit u= limit v; 
but having limit u = limit v, it does not follow that limit — I = 1, 
unless the limit of each is finite and not zero.. 

Theorem VII. The limit of the logarithm of any vai-iable with a 
finite limit, is the logarithm of the limit of the variable. 
Let (i-\-y) represent any variable with a finite limit. 
From Algebra we have 

log (i+;>) = M[y-~ + ~- &c]. 

Hence, Theorem II, 

(limit;') 3 (limit j) 3 
limit log (i+y) =M [limit y— H — &c] = log (1 + limit;'). 

Theorem VIII. Limit a* — a Umit x . 
From Algebra we have 

a x = 1 -f-Cj ^+ c 2 •* 2 + c 3 * 3 +&c., 
in which Cj, c 2 , &c, are constants, respectively, equal to log e <7, 

aog^i &c 

2 ' 

Hence, Theorem II, 

limit a x = 1 -f-Cj limit x+c 2 (limit x) 2 +S:c. —a fimit x . 



PRINCIPLES OF LIMITS. 29 



Theorem IX. Limit sin ip = sin limit ip. 

From Trigonometry we have 

ib s i/> 5 

sin ib=ib — — h T „ „ , . — &c. 

1.2.3 1-2. 3-4-5 

Hence, Theorem II, 

(limit tbY (limit 1b) 5 

limit sin it-' = limit ib — + — &c. sin = sin limit ib. 

1.2.3 1-2.3-4-5 

From the preceeding theorems we learn, that, in general, the limit 
of any continuous function of one or more variables is the same function 
of their respective limits under the law. 

That is, 

limit / (u, v, ....)== /" (limit u, limit v, ....). 

Hence, we have, in general, the following rule for obtaining the 
limit of any continuous function of any number of variables. 

Substitute for each variable its liinit under the law. 

It follows, that those relations which continually exist between 
variables as they approach their respective limits under a law, will exist 
between their limits. 

Theorem X. If unity is the limit of the ratio of any two variables 
with finite limits, the limit of any function of one will be equal to the 
limit of the same function of the other. 

Let u and v represent any two variables, giving limit \-\ =i. 

Then, 

Bmit[/(u)]=limit|/| — I |=/( limit I - I limit v ) = /(limit v) = limit[/(v)]. 



«t[/ (u)]=limit \j\~~\ =/( limit [jjlimit v j 

Exercises. 
Having limit - [ = i, we find, 

Lim. (A±u)=A±lim. u = A±lim. v. 

r(A±u)v~l (A±lim.u)lim.v 
For, lim.(A±u)=lim. [_ — J= ^^ 

=A±lim. - lim.v=A±lim.v. 

Lim. [a u]=Alim. u = Alim. v. 

_ _ tauv"] run 

For, lim.[AU]=lim.| j=Alim. 1 -I lim. V= 

Lim. - =- lim. u=- lim. v. 
LaJ a a 



a lim. v. 



3o 



LIMITS. 



For, lim.g]=lim.[^]^lim. [£] lim. v^lim! v. 

4. Lim. u n =(lim u) n =(lim. v) n . 

ruv~] n rui n 

For, lim. u" = lim. — — = lim. - lim. V n =(lim. v) n . 

5. Lim. -\/u = V lim. u = V lim - v * 

n /— n /^v n /^ n /— n / 

For, lim. V u — ^ m - V ~^~ =lim. V y lim. -y u= y lim. v. 

6. Lim. A u = A lim - u =A lim - v . 

V / /U\ \ lim. V 

For, lim. A u = lim. [" ?1 =( A im ' ^ V ' ) =A lim - v . 

7. Lim. log u = log lim. u = log lim. v. 

For, lim. log u = lim. (log. u — log. V + log v) = lim. log - -Hog v 

— log lim. - +log. lim. v=log lim. v. 

8. Lim. sin u = sin lim. u=sin lim. v. 

uv r /u\ 1 

—-=8111 lim. ( - I lim. v = sin lim. V. 



For, lim. sin u = lim. sin 



.Theorem X enables us to substitute either of two variables for 

the other in any function, without affecting the limit of that function, 

under the law that makes unity the limit of the ratio of the two 

variables interchanged. 

The advantage in so doing arises when we can determine an exact 

expression for one of the variables and not for the other. 

To illustrate, let MM'=s be an arc 

of a plane curve, PM and P'M' the ordi- 

nates of its extremities. Draw the chord 

MM', and denote PP' by ax. Under 

the law that A 3: approaches zero, which 

requires <$■ to approach zero, let it be re- 

• a * \c a r >. rare MM'! 
quired to find limit • 

Having no exact expression for the 
length of the arc MM', it is impossible 




APPLICATIONS. 31 

to find the limit of the above ratio; but it will be shown hereafter that 

limit J u C A f i^r^ = 1 - Hence, Theorem X, 

— ►OLchord J/J/'J 

cos Q' MAI' 

A~~ J 

_ limit r J ~| J 

~~ Sxn->0 L cos Q'MM'J cos Q'MT 

It is important to notice that the above substitution is authorized 
only in taking the limit of a function of the arc, for an arc is never 
equal to its chord. 

From Theorem III, we have limit u = limit v, or - — : — - = 1, 

' ' limit v 

when u — v = d is an infinitesimal; and from Theorem VI, we have 
imit 1 _ ij m i t 5 when u and v have any finite limit except zero. 

Hence, unity is the limit of the ratio of any two variables with finite 
limits, not zero, if their difference is an infinitesimal. 

When each of two variables has zero or infinity as a limit it does 
not follow that the limit of their ratio is unity. 

Let u, v, w, and s, be functions of the same variable, giving 
under a law, limit I - | = i, and limit — =1. 

Then will limit F^]=limit PH. For, Theorem X, limit Prl = 

Umit[X]= limit |J]. 

Applications of the Principles of Limits. 



Limit . 
32 - mn^O I 



Developing I 1 -f- m J m by the binomial formula, we have 

(\ m ,1 1 / 1 \m 2 
/ m m\m / 1.2 ' 

+-L(i-il (i- 2 ) . . . .(±_, !+1 )-J!?— + &c „ 

m \m ) \m J \m J 1.2. ... n 



LIMITS. 



which may be written 



(i+m\ m 



I + I+ I— ^ + (I— ^)(l— 2W) + 



+ &C. 



(I) 



1.2 1.2.3 

(1 — in) (1 — 2m) .... 1 — (11 — i)m 
1.2.3 n 

As m approaches zero, each term in (1) approaches the corre- 
sponding term of the series 

1 



Hence 
lim 



I+I +I^ + 7^ + 



1.2.3. 



%(^'") i 



1 + 1 + — + 

1.2.3 



+ &C. 



+ &c. 



(2) 



I.2.3. 



From Algebra we have 



£=i + i+: 



+ 



+&C.-2.7182818 + 



1.2 1.2.3 1.2.3. • • 

m which e is the base of the Napierian system of logarithms. 



Hence, 



33. 



Limit 



limit / \ m 



In the expression -j— , substitute i-\-y for a x , giving 



.log(i+y)__ 



log 



a x — I y 

Then, since x and y vanish together, 



p+^V- 



limit 



ofe] =;ro [ iog (i+ ^ f] = iog [£&+>)' ] = log 



e=j?/. 



Hence, 



limit I " ••-' 



34. Unity is the limit of the ratio of an angle to its sin, of an 
angle to its tan, and of the tan to the sin, as the angle approaches zero. 

Let OCM—cp be any angle less than 

-; then tan (p><p>sin cp, and 

tan cp cp 

-. > . > 1. 

sin cp sin <p 




APPLICATIONS. S3 

Limit r tano ~|_ limit f i "[ 
<p:~->0 Lsin (pj qm-±Q Lcos <p J 

Hence, * aA i J- 7 ^-~] =1 . 

<£»^-»U|_sin (pj 

Also I> ^->5!V?_ and limit rS*] =I . 

tan (^ tan <£»' <p*-»0|_tan <pj 

Hence, lirait T-?-~] =I . 
<£>pe— »Ol_tan <£>J 

T limit rsin- 1 ^ "1 

Let *=sin<p, ,\ ^sin- 1 *, and ^^_^ [_ ^ J = L 

Let z/=tan<p, .'. <p=ta.n- 1 tt, and MB _^ |_ — " — J=i. 

Similarly, it may be shown that unity is the limit of the ratio of 
each pair of the lines PM, OT, and M\ as M approaches zero. 

35. Let s be an arc of any 
curve of double curvature, AC a 
tangent to it at A, AB the chord 
corresponding to s ; and s' the 
projection of the curve upon the 
plane of the tangent AC and chord 
AB. AC will also be tangent to s f 
at A. [Des. Geo.]. 

Assume any number of points upon s, including A and B, and 
connect adjacent ones by right lines. Represent the chords of s } 
thus formed, by c, c', etc. 

Let 6, 6 f etc., denote the c^sts^^^^fee angles made by the 
chords respectively with the plane CAB. 

The projections of the chords c, c' , etc., upon the plane CAB 
will be chords of $'. [Des. Geo.]. 

Let the points be taken nearer and nearer to each other, and let 
// denote the number of chords; then 




. limit /,_l^#_ i _a«.V— limit V*. «« a ,/— limit v 



i=^T^-K l +fc.)=™^; ^d ,'= ™ 2<cos5. 



Under the law 5»»-*0, the cos of each of the angles 6 #', etc., 
will approach unity. Hence, §31, Theorem X, 



[nnm ^ -1 
limit v " 



*.2 is used to denote the sum of any number of terms similar in form to the 
one written after it. 



34 



LIMITS. 




36. Unity is the limit of the ratio of an arc of any cw've to its 
chord, as the arc approaches zero. 

Let RT=s be an arc of any 
plane curve, assumed so small 
that it is concave throughout to- 
wards its chord w. 

Draw the tangents RW=t, 
and TW—r, completing the tri- 
angle R WT; and from W draw WP perpendicular to w. 
As s approaches zero, we have 

t-\-r>s>w, and w=t cos R-\-r cos T; 
and since the angles R and T also approach zero, 

limit p-Kn limit r *+ r 1 

Hence, limit -I"— l = i. 

In case the given curve is one of double curvature, the tangent 
R W will not, in general, intersect the tangent at "T. Project the 
curve, and the tangent at T, upon the plane of RT and RW. Let 
s and TWbt their respective projections. 

From 8 35, limit J— c ]=i 

Hence, § 31, Theorem X, 



limit r^rc - 
arc m-^> L w 



i = limit m 



37. Let MM'=s be 
any arc of a plane curve? 
PM and P' M' the ordi- 
nates of its extremities. 

Through M draw the 
chord MM'~c, the tangent 
MT=b, and MQ' = PP' = 
/\x, parallel to X. 

From the triangle MM'T, we have - = s | n ZZ'J, . 

As Ai approaches zero, the arc s and the angle M ' M T will 
also approach zero, but the angle T will remain constant. Hence, 
the angle MM'T will approach [180°— T], and we have 
limit ril— limit pinJlffl/' 7 *1 sin (180 — 7») 
0|_ sin r J sin T 



y 


ft/ 


M' 






N/ 


^ 




A2/ 

if' 

p' 




/ 




1/ 
P 


Ax 


X 



t fi] = 1 



=1; 



and since, 



b h 

7>s>*> 



APPLICATIONS 

limit 



35 



we have 



$m-+ 



£]•■ 



From the same figure we have 
Hence, § 31, Theorem X, and § 36, 



Ax 



"cos Q'M T 



r Ax -1 

limit [.=£_"] = limit cos^M/r = I 

,xsb>-»0l. AxJ Axb->0 cos Q'MT 

i_ A^" —J 




> or 



38. Let BMM' be any plane curve, and 
PMM'P' the plane surface included between y 
any arc of the curve, as MM', the ordinates of 
its extremities, and the axis of X. 

Through M and M\ respectively, draw MQ' b 
and M' Q parallel to X, and complete the rect- 
angle MQM'Q'. 

Let PP'—Iax approach zero. Then, since 

PQM [P ' > PMM 'P' > PMQ'P', and 

we have limit \ PMM ' P ' ~\ ' 
ve toe ax^oL pmq'p>]= 1 - 

Hence, § 31, Theorem X, 

limit r PMM'P' i limit r PMQ'P 'i limit [ *** ! 



limit r PQM'P' -} 
Axvh+QL PMQ'P' J~ 



If the coordinate axes make an angle 



limit 



with each other, then 

sin h Ax' 



it r PMM'Pn i^it psinS Ax l . 



39. Unity is the limit of the ratio of the surface of revolution 
generated by any plane arc, revolving about an axis in its own plane, 
to that generated by its chord, as the arc approaches zero. 

Let PT=s be any plane arc 
in the plane XY, and PP=y, 
and QT=y', the ordinates of its 
extremities. 

Draw the chord RT—w y 
and the tangents PlV=t, and 
TJF=r, forming the triangle 
RWT. Draw WP"—y n perpendicular to X. 




3<> 



LIMITS. 



Let the figure be revolved about X; then 

sur. gen. by t — 27ti ~7~) t = tc (y+y rr ) t. 

/ y '+y \ 

sur. gen. by r— in\ — - — I r = it (y"-\-y') r. 

(y+y'\ 
sur. gen. by zv = 27f( Jw ■= % (y+y')tc;. 

Under the law that s approaches zero, we have limit y' dimity" =y. 
Hence, 

limit i / , , \ limit . / , «\ . , limit _ , „ . /N limit r / . . ;-i 

; ;>^-»osur. gen. by (/+r)= s »->o?r (j+jj/ ) /+ S )»^-»otf (j/ +y) r= s ^-»oL2 Tt y \t+r)\. 

limit , limit _ / , /\ limit r -, 

,\*-»osur. gen. by w = S )»>->oit-{y+y) w= B m-^ol2 7eyw\. 



Hence, 



Since 



limit pur, gen, by (/+r) ~\_ limit p + r~l „ , 

^^->0l sur. gen. by w J .r >»»-> L f« J \ ' 



sur. gen. by (t+r) sur. gen. by s 
sur. gen. by w sur. gen. by w ' 



limit pur, gen, by s 1 
•B-> OLsur. gen. by zvj ' 



Hence, § 31, Theorem X, 

T 




P Ax 



limit r sur.gen.byarcyFAr i 
X3»-»0L Ax J 

limit r sur.gen.bych.^/^ /^l 

r it(y+y)e l 

OL Ax J 

n ^ + ^co S Q'MM'] = 27r -> 
—J COS (2'i 7 



Ax»»-»oL Ax 

limit r *(j'+y)* 

Ax 

limit 

AxB->0 



Ax 



y r 
MT 



40. Let BMM' be any plane curve, and 

PMM' P' the plane figure bounded by any arc, 

M' as MM' , the ordinates of its extremities, and the 

^ axis of X. Through M and M', respectively, 

Q' draw if (2' and if' <2 parallel to X, and complete 

the rectangle MQM'Q*. 
£— ^ Let the entire figure be revolved about X, then 



vol. gen. by PQM'P'>\o\. gen. by PMM'P'y vol. gen. by PMQ'P'; 



APPLICATIONS. 



37 



and since 

]imit r vol.gen.by/'QJ/'/" 



_i.v m-*0 



r voi. 

Lvol. 



gen. by FMQ 'P 



i , therefore 

limit 
Ajcs&h 



[" vol. gen, by PMM'P' l 
L vol. gen. by FMQ 'P ' J ~ 



Hence, § 31, Theorem X, 

limit r v ol.gen.by^J/J/'^- |_ Umit r vol. gen. by/M/g'^ - 

1/2 



# 



Ax 



limit 



41. Let r=f(z>) be the polar 
equation of any plane curve, as 
AAfM', referred to the right line 
PZ>, and pole P. 

Let AM=s be any portion 
of the curve, and PM—r the 
radius vector corresponding to M. 

Regarding s as a function of 
r\ ij 19, let v be increased by 



P£- ]= ., 



J//W 



AZ>. 



The arc J/J/' 



will be the corresponding incre- 
ment of s. Draw MQ' perpen- 
dicular to PM', and denote PM f by r' 
and § 36, we have 




Then, § 31, Theorem X, 



limit arc MM' I limit ch. MM' limit / MQ'^+Q'M' 



limit 



\/^°^ 



) 3 +(/'' — rcosAt/) : 



limit 
A^^-> 



Also, 



limit A / />' — r\ 2 

limit r , nfjtffjLii limit r ^ /yT/ ~1— limit f rsinAz; 1 



AfB-> 



_ limit 



0|_r'— rj - 



la.nPMD. 



If the radius vector /W coincides with the normal to the curve 
at M. we have angle PMT=PMB-- 



J^q e'™=90< 



38 



LIMITS. 



Since MT> arc MM'>-MQ', and 
MT 



limit 



limit 



Lsin Q'TMJ 



we have limit n 

A^»»->0 



Hence, 



limit 



\MT_~\_ 
OlMQ'J 

Tare MM'~\ 
L MQ> i = I - 

r arc A/M' l H mit fMQ'l u 

L AP J A v M-> L A v J A & 



limit 



T r sin A ^ 1 



42. Let MPM' be the surface 
generated by the radius vector PM=r, 
revolving about P, as a pole, from any 
assumed position, as PM, to any other, as 
PM f . Let Av represent the corre- 
sponding angle MPM' . With ? asa 
centre, and the radii PM and PM', 
describe the arcs MQ' and M' R respect- 
ively. 
Then, since area RPM '> area J// 5 M' > area J/i^' 

limit r areaA'/W l , limit f area Jf/>Jf' 

A z/ s»-> L area MPQ' 

Therefore, 




and 



: ]= 



we have 



limit I area^/^/' -| 
Az/^^-OLarea J/7>()' J 



limit r area^PJ/"' -! _ limit r area JiPQ' l _ limit 
, v »-» L A v J A z/ M-> L A z> ! A v B-» 



,-2 



AW 



43. Let DABCF be a plane 
figure, and DA' B'C F its projection 
on another plane intersecting the first 
in the right line DF. 

Assume any number of points on 
DF, through which draw right lines 
parallel to DA and DA' respectively. 
Through the points in which the 
first set intersect the curve ABC, and 
the points in which the second set inter- 
sect A' B'C, draw right lines parallel to DF, forming the two sets 
of parallelograms AE, BF, etc., and A'F, B'F, etc. 

Through A A ', the projecting line of A, pass a plane perpen- 
dicular to DF cutting the two planes in the right lines AP and 
PA' respectively. 




APPLICATIONS. 



39 



The angle A PA' ', which we will denote by 6, is that made by 
the planes with each other. 

A'P=APcos f J. areaAE=APXE>£. are2iA'E=A'PxEE=APcos0xEE 

=area.AEcos f ). 

Similarly, each parallelogram in projection is equal to the corre- 
sponding one in the given figure into cos 6. 

Let n denote the number of parallelograms, then as the number 
of points on DF is increased we have 

EA'P'C'P=n ] ^ t x (A'£+B'E+Scc.)= I1 ^ x (A£+BP+kc.)cosO 

=DABCP cos b. 

In a similar manner it may be shown that the area of the pro- 
jection of any plane figure, is equal to that of the given figure into the 
cosine of the angle made by its plane with the plane of projection. 



44. Let MNM'N' be 
a portion of any surface, in- 
cluded between the coordi- 
nate planes ZX, ZY, and the 
two planes N'SE and NPD 
parallel to them respectively. 

Let OP—Ju and OS—k. 

At M draw the tangents 
MB and MB' to the curves 
MN and MN' respectively, 
and complete the parallelo- 
gram MBQB'. It will be the 
portion of the tangent plane 

to the surface at M, included between the planes which limit 
MNM'N'. Draw the chords MN, NM', M'N', and N'M, 
forming the quadrilateral MNM' N' inscribed in the assumed curved 
quadrilateral MNM'N' . Draw the diagonals MM', MQ and OP. 

Conceive the concave surface of the curved triangle MNM' to 
be entirely covered with inscribed plane triangles, formed by assum- 
ing a sufficient number of points, including M, N, and M' on the 
surface, and connecting those adjacent by right lines. 

Let /, t', /", etc., represent the areas of the triangles respect- 
ively; and let 6, 6', 6", etc., represent the angles made by their 




40 LIMITS. 

respective planes with the plane of the plane triangle MNM'. 
Then, P^ne triangle MNM ' =t cos + t' cos ft' + t" cos 0" + &c. ; 

or, denoting the sum of all the terms in second member by 2 1 cos 6, 
we have plane triangle MJVM'=2 t cos ft. 

Denoting the number of the inscribed triangles by «, and in- 
creasing them indefinitely, we have 

curved triangle M2VM'==Sf<* 2 1, 
plane triangle MNM'= n ^^^ 2 1 cos 6. 

Suppose h and k to approach 0; or what is equivalent, let OF, 
represented by /, approach 0. The plane triangle MNM' will 
approach coincidence with the tangent plane at M. Each of the 
angles 6, 6', etc., will approach 0, and for each we have 

limit [-11 
/»»-^oLcosJ 

Hence, § 31, Theorem X, 

/£* [plane triangle *Vif']=/^ [»S» . 2*], 

and 

^[curved triangle MArM^^^^Z t} 

Therefore, 

[limit -ST. ,~"l 
n ft°° =1 

From § 37, 

limit ( M Q\ limit (^\_ A limit V an g le BM Q 1 

lT*h+0\MM' J~ r » lw^0\MNj~ lf and /^OL angle ,VJf;¥' J -1 - 

Hence, 

limit f tri. MBQ -1 limit f tri. JfgQ ~| 

/^-^0L plane tri. J/7W J -1 ' ana /^0|_curved tri. MNM'}- 1 ' 

In a similar manner, it may be shown that 

limit f trlMB'Q 1 
/^-» OLcurved tri. ifiV 'if' J 
Hence, 

limit r quadrilateral MBQB' ~\ 

/l /^—^ 01 I zzz T 

A;B->oLcurved quad. MNM'N'A 

45. The volume of MNM'N'—OF, included between the 
coordinate planes, the two planes N' SE and NPD, parallel 
respectively, to XZ and YZ, and the curved surface MNM'N', 



APPLICATIONS. 



41 



is greater than that of the parallelopipedon OPFS—M'C, and less 
than that of OPFS—MB. 




Let OP=/t, and OS=k, approach zero. Then, since 

vol. OPF S—MB I 

>— M'CJ- 1 ' 



op 

oLv 



limit 
h fib — ^ 
*s»-X)Lvol. OPFS- 



limit fvol. MNM'N'—OF~~\ 



we have A^o ^p^c — 77^- I=i. 

>0L vol. OPFS—MB J 

Hence, § 31, Theorem X, denoting the ordinate MO by 2, 

limit i-vol. MNM'A T, —OF~ 



limit r-i 



/; £ 



] limit ["vol. 
— ^B-^o 
£»»->0L. 



OPFS— MB 
hk 



~i limit r z kk~\ 
\=hfi»->0\ — — 



CHAPTER III. 

RATE OF CHANGE OF A FUNCTION. 

46. In the function 2x 2 , a change in the variable from 2 to 3, 
causes the function to change from 8 to 18. If x be again increased 
the same amount, that is from 3 to 4, the function will increase from 
18 to 32. Similarly, with other functions we shall find that, in general, 
equal changes in the variables do not give equal changes in the 
corresponding functions. 

It is therefore necessary, in referring to a change in a function 
corresponding to a change in the variable, to consider the states from 
which and to which the function and variable change, as well as the 
amount of change in each. With that understanding, coi-responding 
changes in a function and its variable are mutually dependent. 

Thus, having u=f(x) .... (1), hence, § 4, x—F{ii) .... (2), 
increase any value of x in (1) by /z, and let k denote the corre- 
sponding increment of the function u. Now if the variable u in (2) 
be increased by k from the state that u in (1) had for the first value 
of x; the function x in (2) will change by h from and to the same 
values that the variable x in (1) had. 

47. A function changes uniformly with respect to a variable, 
when the ratio of any two increments of the variable is equal to that 
of the corresponding increments of the function. 

It follows that any equal increments of such functions will corre- 
spond to equal increments of the variable. 

Thus, in 2ax, let h and / represent any two increments of the 
variable x. The corresponding increments of the function are 
2ah and 2al. 



h 2ah 

1 2a/' 

Hence, 2ax changes uniformly with respect to x. 



f — —j, and if &=/, then 2a/i=2al. 



RATE OF CHANGE OF A FUNCTION. 



43 



/ 



c 


c 




B 


/ 


Q 

$ 

p" 


A 


/ 


Q # 




^ 


P 


P' 


p'" X 



To illustrate, the function 2<zx 
will be represented by the ordinate 
of some right line, as ABC. In- 
crease any value of x, as OP, by 
any value, as PP'=h. Q'B will 
be the corresponding increment of 
the function. Then increase x=OP 
by any other value, as PP"=/, 
giving the increment S"C to the 
function. The similar triangles 

AQ'B and AS"C, give ~ = §r§- 

Hence, the ordinate of 

function, changes uniformly with x. 

By giving to x=OP-, any equal increments, as PP', P'P" P"P'", 
in succession, the corresponding increments of the function, Q'B, 
Q"C, and Q" r D, are equal to each other. 

In a similar manner, it may be shown that any function, which is 
represented geometrically by the ordinate of a right line, changes 
uniformly with its variable. 

Any function which is of the first degree with respect to the 
variable, is some particular case of the general form Ax-\-B, in 
which A and B are constants. 

Such functions are represented geometrically by the ordinates of 
right lines, and will change uniformly with their variables. 



the right line 



ABC, and therefore the 



48. In the function 2xj x=i, gives 2x= 2. 

x=2, gives 2jc= 4. 

x—$, gives 2x— 6. 

From which we see that the function increases two units while 

the variable increases one; in other words, twice as fast. 

Having $x; x=i, gives s x = 5- 

x=2, gives 5>r=:io. 
x=3, gives 5^=15. 
Which shows that the function changes five times faster than the 
variable. 

Hence, different functions, in general, change with their variables 
with different degrees of rapidity. 



44 RATE OF CHANGE OF A FUNCTION. 

The measure of the relative degrees of rapidity of change of a 
function and its variable at a?iy state, is called the rate of change of 
the function, with respect to the variable, corresponding to the state. ' 

A rate of change of a function with respect to a variable, corre- 
sponding to a state, is an answer to the question: At the state con- 
sidered, how many times faster than the variable, is the function 
changing ? 

49. Since any function, which changes uniformly, receives equal 
increments for any equal increments of the variable; it follows that 
the rates of change of such a function, corresponding to different 
states, must be equal; for otherwise, the function would receive 
greater or less increments for equal increments of the variable. 
Hence, the rate of any function, which changemniformly with respect 
to a variable, is constant. 

50. From the definitions of uniform change and rate, it follows 
that the rate of a function which changes uniformly with respect to 
a variable, is equal to the ratio of any increment of the function to 
the corresponding increment of the variable. 

Thus, having any f(x), which is of the first degree with respect 
to x, increase x by any convenient increment h. f(x-\-h)—f(x) 
will be the corresponding increment of the function, and the rate 

will be ^ j . This ratio is independent of h, hence // may 

be made zero without affecting the rate. 

™, . r „ 2(x-\-/l) — 2X 

Thus, rate of 2x—^ J. _ 2 

h 

Rate of 3-r+^ [3( " + * )+ * ] - [3x+2] = 3 
Rate of iJ - 3= b' I + / 'HHs>-3] 

h D ' 

It follows, that the product of the rate of a function, which 
changes uniformly, and any increment of the variable, is the corre- 
sponding increment of the function. 

51. In the function ax 2 , let h and / represent any two in- 
crements of x. The corresponding increments of the function are 
2axh-\-ah 2 , and 2axl~\-al 2 . 

The ratio - is not, in general, equal to ax \ a * ; hence the 

/ e>- ■.. . i s i 2axl+al* ' 

function ax 2 does not vary uniformly with x. 



RATE OF CHANGE OF A FUNCTION. 



45 



In a similar manner it may be shown that any function, which is 
not of the first degree with respect to a variable, does not change 
uniformly with the variable. 

To illustrate, take any function of a 
degree higher than the first. It will be 
represented by the ordinate of some curve, 
as MNT. Increase any value of x, as 
OP', by P'P", and P'R; then, since 

^ = , , the ratio ^-, of the correspond- 



ing increments of the function is not, 
P'P" 



in 













T 




Y 




y 


// 








B 


N 









u 


// 




Q 




S 




p' 


P" 


R X 



general, equal to 



P'R 



52. In the function 2X 2 : 



2XT—2. 



X=2, glVeS 2X' 



gives 



<6 



<io 



<I4 



x=4, gives 2^=32. 

Which shows that at different states the function 2X 2 has differ- 
ent rates with respect to x. 

Similarly, it may be shown that any function which does not 
change uniformly has, in general, different rates at different states. 
In other words, the rate varies with the function and its variable. 
Any particular rate is, therefore, designated as the rate correspond- 
ing to a particular state. 

If a function has two or more states corresponding to any value 
of the variable, each state will have a rate. 

If a function has equal states for different values of the variable, 
it may have a different rate at each ; in which case it is necessary 
to indicate the value of the variable corresponding to the state 
considered. 



53. Let P(x) be any function which does not change 
uniformly with its variable. Denote its rate, corresponding to any 
particular state, by P. Increase the corresponding value of x by 



46 RATE OF CHANGE OF A FUNCTION. 

h, and let R' represent the rate of the function at the new state 
F(x-\-Ji). Let h be taken so small that the rates between, and 
including R and R', shall increase or decrease in order. 
F(x-\-/i) — F(x) will be the corresponding increment of the function. 

The ratio — - — — is not the rate R, for the change 

F(x-\-h) — F{x) is due to all rates from R to R'; but the ratio 

— — ~ multiplied by h gives the increment F(x-\-h) — F(x); 

hence, the ratio — - — — ~ is the rate of another functio?i of x, 

which varying uniformly between the states considered, will change 
by an amount equal to that of the given function, 

To illustrate, let the given function 

be the one represented by the ordinate 

of the curve AB. Let FA represent 

the state at which the rate is Rj and 

let PP'—h be the increment of the 

variable. P'B will then represent the 

state at which the rate is R', and QB 

will be the increment of the function 

corresponding to h. 

Draw the right line AB. Its ordinate will represent a function 

which changes uniformly from the state PA to P'B, and by an 

amount QB equal to that of the given function. Therefore, 

QB=F(x+h)-F{x), and QB = F^+k)-F(,) . but 0^ and there . 

fore — - — — , is the constant rate of the function represented 

by the ordinate of the right line AB. 

The constant rate of the function represented by the ordinate 
of the right line must be greater than the least, and less than the 
greatest rate of the given function for the states under considera- 
tion ; otherwise the function represented by the ordinate of the 
right line would change by a less or greater amount than the given 
function between the states considered. Hence, we have either 

F(x+k)—F(x) F(x+h)—F{x) 
K< — 1 — <R' t or R> — 1 ^ >R'; 




RATE OF CHANGE OF A FUNCTION. 



47 



depending upon whether the rates from R to R' are increasing or 
decreasing. 

One or the other of the above relations will exist always as h is 
diminished numerically; and since, in either case, R is the limit of 
R' under the law that h approaches zero, we have 

limit r F(x+A)-F(x) l 
/m-*0 L h J — K ' 

That is, the rate of change of any function with respect to a 
variable, corresponding to any state, is equal to the limit of the ratio of 
any increment of the function, from the state considered, to the corres- 
ponding increment of the variable, under the law that the increment of 
the variable approaches zero. 

The above principle enables us to find the rate of any function 
with respect to a variable, corresponding to any state, by the follow- 
ing general rule. 

Give to the variable a?iy variable increment, and from the corres- 
ponding state of the f miction subtract the primitive. Divide the 
remainder by the increment of the variable, and determine the limit of 
this ratio, under the law that the incre?nent of the variable approaches 
zero. In the result substitute the value of the variable corresponding to 
the state. 

It should be observed, that a rate, determined by the above 
method, is equal to a limit of a ratio of two infinitesimals, which 
limit is determin ate ; and that it is not equal to the ratio of their 
limits, which ratio is -§-, and therefore indeterminate. See § 31, 
Theorem VI. 



54. To illustrate the changes 
which occur in the ratio of the 
increment of the function to that of 
the variable under the above law; let 
the given function be represented 
by the ordinate of the curve AB" B f , 
and let PA be the state considered. 

The ratio, for h=PP', is ^ , 
which is the rate of the function 
represented by the ordinate of the 
right line AB'. 




48 RATE OF CHANGE OF A FUNCTION. 

D" R" 

For h=PP", the ratio ^p-^r , is the rate of the function repre- 
sented by the ordinate of the right line AB" . 

As h is diminished, the ratio is always the rate of a function 
represented by the ordinate of a secant approaching the tangent 
A T; and the limit of the ratio is the rate of the function represented 
by the ordinate of the tangent A T. 

That is, the rate of the given function, at the state PA, is the 
same as that of a uniformly varying function represented by the 
ordinate of the tangent A T. 

This is consistent with previous conceptions and definitions, for 
the direction of the motion of the point, generating the curve at 
any position, is along the tangent at the point ; and the rate of 
change of the corresponding ordinate of the curve and tangent, 
must be the same. 

55. § 31, Theorem VI, 3 , shows that the limit of the ratio of 
any increment of a function from any state, to the corresponding 
increment of the variable, under the above law, is equal to the 
tangent of the angle made, with the axis of abscissas, by a tangent 
to the curve, whose ordinate represents the function, at the point 
corresponding to the state considered. Hence, the rate of a 
function with respect to a variable at any state, is equal to the tangent 
of the angle above described. 

Exercises. 
Find the rate of change of each of the following functions. 
limit [ 2a{x+k)—2ax ~] 

r - 2ax - Ans - /^oL h \ =2a - 

limit [- (*+A)«-*M o 

2. x\ Ans. ^^ ~ h \=ix. 

limit r a(x+A)*+5(x+A)-(ax* + bx) ~\ ' 

3. ax* + 6x. Ans. ^^ [_ ~ h J =2ax+&. 



a limit 

4- ~- Ans. ^^0 

5. 2ax 2 . Ans. $ax. 

6. x s . Ans. 3x' 2 . 

7. 4^r 4 . Ans. 16* 3 . 



a a 

x-\-h x 



EXERCISES. 49 



l+x ■ AnS ' (l+^) s ' 

2x 6 



Q. — — . Ans. 

10. How is the ordinate of a parabola, corresponding to x=$, 
changing with respect to the abscissa ? 

j . ; t limit f V22J(x+h)— V2^x ~\ 

y=V2 P x, . . rate of y= ; ^ [_ % J 

(VD^Vf An, 

ii. Same corresponding to focus ? Ans. i. 

12. Find the abscissa of the point, of the parabola y 2 =4x, where 
the ordinate is changing twice as fast as the abscissa. 



Rate of y—i .'. 2= \/ 2x = \/ ^ 



2^ 

2x 



4 

13. At the vertex of a parabola, how is the ordinate changing as 

compared with the abscissa ? 

14. Find the rate of change of the abscissa of a parabola with 

respect to the ordinate. 

y _ A /2x m 
Ans. - - V - • 

15. Find the coordinates of the point of the parabola y 2 =Sx, where 

the abscissa is changing twice as fast as the ordinate. 

p y Ans. y=S. 

~ y ~ 4 x=S. 

16. Find the rate of change of the ordinate of the right line 

2 y— 3#=i2, with respect to the abscissa. 

Ans. 

2 

17. A point moves from the origin so that y always increases 

I times as fast as xj find the equation of the line generated. 

- = tan of angle line makes with X. .'. Ans. 4.y=$x. 



50 RATE OF CHANGE OF A FUNCTION. 

56. Motion.* When a point changes its position with respect 
to any origin it is said to be in motion with respect to that origin. 

In general, the distance trom any origin to a point in motion 
continually changes, and is a continuous function of the time during 
which the point moves. 

When the distance changes so that any two increments of it 
whatever are proportional to the corresponding intervals of time, 
the distance changes uniformly with the time. The point is then 
said to be moving uniformly, or with uniform motion with respect 
to the origin. 

If the distance does not change uniformly with the time the 
point is said to be moving with varied motion with respect to the 
origin. 

A train of cars moves from a station with varied motion until it 
attains its greatest speed, after which its motion along the track is 
uniform while it maintains that speed. 

With uniform motion equal distances are passed over in any equal 
portions of time, and with varied motion unequal distances are passed 
over in equal portions of time. 

Let s represent the 

A s M. N O variable distance from any 

origin as A, to a point 
moving on any line, as 
MNO; and let t denote 
the number of units of 
time during which the point 
moves; then s=f(t). 

If /(/) is of the first 
degree with respect to t, 
the distance s will change uniformly; otherwise the point approaches, 
or recedes from the origin with varied motion, § 47, § 51. 

The rate of change of s, regarded as a function of t, corres- 
ponding to any position of the moving point, is called the rate of 
motion of the moving point with respect to the origin; and since 

*Motion, without regard to cause, is generally discussed under the head of 
Kinematics, but many important applications of the Calculus involve motion, there- 
fore, some of the definitions and principles of Kinematics are here and elsewhere 
introduced. 





RATE OF CHANGE OF A FUNCTION. 5 1 

uniform motion causes s to change uniformly with f, the rate of 
motion, in such cases, is constant. § 49. 

In varied motion, the rate varies with /, and is therefore a 
function of /. 

Let C be a fixed point, CA a 
fixed right line, and B a point in 
motion so that the angle ACB, 
denoted by 6, is changing. Then 
the line CB is said to have an 
angular motion with respect to, or 
about, C. 

Let s represent the length of the varying arc, of any convenient 

circle, subtending 6, giving 6= — . 

Both 6 and s are functions of the time during which CB moves. 

Angular motion is nnifoi-m when any two increments of the 
angle, or arc subtending the angle, are proportional to the corres- 
ponding intervals of time; otherwise it is varied. 

57. A function of two variables changes uniformly with respect 
to both variables when it receives equal increments corresponding 
to any equal increments of each variable. 

Every function of two variables, which is of the first degree 
with respect to the variables, must be some particular case of the 
general form Ax-\-By-\-C, in which A, B and C are constants. 

Placing z=Ax-{-By-\-C, and increasing x by h, and y by k y 
we have for a second state z'=A(x-\-h)-\-B(y-\-k)-\-C. 

Again increasing x by //, and y by k, we have for a third state 
z"=A(x+2/i)+B(y+2k) + C. 

z f — z=A/i-\-B&, is the increment of the function from the 
primitive to the second state. 

z" — z' = A/i-\-Bk, is the increment from the second to the third 
state. 

These increments of the function are equal, and correspond to 
any equal increments of each variable. Hence, any function of two 
variables, which is of the first, degree with respect to the variables, 
changes uniformly with respect to both variables. 



52 RATE OF CHANGE OF A FUNCTION. 

58. Let 2=/(x, y) = Ax 2 -\-By-\-C. Increase the variables, 
respectively, by h and k, giving the new states, 

z'=A (x+/i) 2 +B(y+£) + C, and z"=A{x+2h)*+B (y+2k) + C. 

Hence, 

z'—z=z 2 Axh-\-Ah 2 -\-Bk, and z"— z' =2 Ax/1+3 Ah 2 +Bk. 

The increments of the function, corresponding to equal incre- 
ments of each variable, are unequal, hence the function does not 
change uniformly with respect to both variables. 

In a similar manner, it may be shown that any function of two 
variables, which is not of the first degree with respect to the variables, 
does not change uniformly with respect to both variables. 

59. Any function of two variables which changes uniformly 
with respect to both variables must be of the first degree with respect 
to the variables, and its form must be some particular case of the 
general expression, Ax-\-By-\-C. 

It also follows, that the surface, whose ordinate represents a 
function of two variables which changes uniformly with both vari- 
ables, is a plane. 

60. In a similar manner, it may be shown that any function of 
any number of variables, which changes uniformly with respect to 
all the variables, must be of the first degree with respect to the 
variables. 

6 1 . The Calculus is that branch of mathematics by which measure- 
ments, relations, and properties of functions are determined from 
their rates of change. 

It is generally divided into two parts. 

Part I, called Differential Calculus, embraces the deductions and 
uses of the rates of functions. 

Part II, called Integral Calculus, treats primarily of methods for 
determining functions from their rates. 



CHAPTER IV. 

THE DIFFERENTIAL AND DIFFERENTIAL COEFFICIENT 
OF A FUNCTION. 

62. An arbitrary amount of change assumed for the independ- 
ent variable is called the differential of the variable. 

It is represented by writing the letter d before the symbol for 
the variable ; thus dx, read " differential of x" denotes the differ- 
ential of x. 

It is always assumed as positive, and remains constant through- 
out the same discussion unless otherwise stated. 

63. The differential of a function of a single variable is the 
change that the function would undergo from any state, were it to 
retain its rate at that state, while the variable changed by its differential. 

The differential of a function which varies uniformly with its 
variable, is the change in the function corresponding to that 
assumed for the variable. 

To illustrate, let PA be any state of the 
uniformly varying function represented by 
the ordinate of the right line AB. Assume 
PR=dx. 

QB, the corresponding change in the 
function, is the differential of the function. 

The differential of a function which does not vary uniformly 
with its variable, is not, in general, the corresponding change in the 
function; but it is the corresponding change of a function having a 
constant rate equal to that of the given functiop at the state con- 
sidered: or, in other words, it is the change that the function would 
undergo, were it to continue to change from any state, as it is 
changing at that state, uniformly with a change in the variable equal 
to its differential. 




54 



DIFFERENTIAL OF A FUNCTION. 



Y 




M 


A 




Q 

R X 





P dx 



To illustrate, let PA be any state of 
a given function represented by the ordi- 
nate of the curve AM. Assume PP=dx. 
QM is the corresponding change in 
the function; but QB, the correspond- 
ing change in the function represented 
by the ordinate of the right line AB 
drawn tangent to AM at A, is the differ- 
ential of the given function corresponding 
to the state PA. For the function represented by the ordinate of 
AB has a constant rate equal to that of the given function at PA, 
§ 54; and QB is the change that the given function would undergo; 
were it to continue to change from the state PA, as it is changing 
at that state, uniformly with a change in x equal to dx. 

The differential of a function 
which does not vary uniformly with 
its variable, may be less than the 
corresponding change in the function. 
Thus, QB, < QM, is the differential 
of the function represented by the 
ordinate of the curve AM, corre- 
sponding to PA. 

A train of cars in motion affords a familiar example of a differ- 
ential of a function. 




B 



Suppose that a train of cars starts from the station A, and 
moves in the direction A E with a continuously increasing speed. 
Let x denote the variable distance of the train from A at any 
instant; it will be a function of the time, represented by t, during 
which the train has moved, giving x=/(t). 

Suppose the train to have arrived at B, for which point x=AB. 
Let B D represent the distance that the train will actually run in 
the next unit of time, say one second, with its rate constantly 
increasing. 

Let B C represent the distance that the train would run, if it 
Were to move from B with its rate at that point unchanged, in a 



DIFFERENTIAL COEFFICIENT OF A FUNCTION. 55 

second. Then will the distance BC represent the differential of x 
regarded as a function of t, corresponding to the state x=A£; and 
one second will be the differential of the variable. 

The differential of a function is denoted by writing the letter d 
before the function or its symbol. 

Thus, d2ax 3 , read " differential of 2 ax 3 ," indicates the differ- 
ential of the function 2ax?. 

Having J— log^/^, we write dy=d\og^/7ix*. 

~- dx denotes the differential of y regarded as a function of xj 

and —j-dy is a symbol for the differential of the inverse function; 
that is, of x regarded as a function of y. 

64. From the definition of a differential of a function, and 
from § 50, it follows, that a differential of a function is the product 
of two factors ; one of which is the rate of change of the function 
at the state considered, and the other is the assumed differential of 
the variable. Hence, the differential of any given function may be 
determined by finding its rate, by the general rule, § 53, and multi- 
plying it by the differential of the variable. Thus, having the 
function 2x 2 , we find, § 53, 

p(.r+/fr) 8 — ^ 2 - 

4xdx is, therefore, a general expression for the differential of 
2x 2 , and is written d2x 2 =4xdx. 

Its value corresponding to any particular state is obtained by 
substituting the value of the variable corresponding to the state; 
thus, for x=2, we have d2x 2 =8dx. 

65. Since the rate of change of a function is the coefficient of 
the differential of the variable, in the expression for the differential 
of the function; writers on the Calculus have, in general, adopted 
for it the name "differential coefficient." 

The differential of a function is therefore equal to the product 
of the differential coefficient by the differential of the variable. 

It follows, that the differential coefficient is the quotient of the 
differential of the function bv the differential of the variable. Thus, 



-^— — 1 — =4^= rate corresponding to any state. 



56 DIFFERENTIAL COEFFICIENT OF A FUNCTION. 

having d2x 2 =4xdx, the differential coefficient is — — =z^x j or, 
having denoted any function of x, by y, and its differential by dy, 
its differential coefficient is represented by ~- . 

The differential coefficient of any function of a single variable 
may be determined by the general rule, § 53. 

Thus, having y=f(x), in which y represents any function, of 
any variable x, let y' denote the new state of the function corres- 
ponding to the increment h of the variable. Then, 

limit [" /(*+ ^)—/(*) 1 _l limit \yjZl\ -u. dy_ . 
7/m->0 L h J /im->0 L h J dx J 

or, representing the increment of x by Ax, and that of y by Aj', 
we have 

limit [AJT a>_ 

Since the increment of the variable, represented by h or Ax, 
varies, it may happen that /z= Ax = dx. It is exceedingly important 
to observe, however, that the corresponding value of y' — y or a^, 
is not, in general, equal to dy j for that would give' 

fy'—y\ =(^l\ i = d -l ■ 

\ /z J h=dx \AxJ t^x-dx dx J 

which, in general, is impossible, since -~ is not a value of the ratio 
, . but is its limit under the law that h vanishes. 

h 

If, however, the function changes uniformly with respect to the 
variable, y ~ y will be constant for all values of h, § 50; and y f — y 
will be equal to dy when h is equal to dx. 

66. The following important facts in regard to a differential 
coefficient should now suggest themselves to the student. 

It is zero for a constant quantity. In other words, a constant 
has no differential coefficient. 

It is constant for any function which varies uniformly. 

It varies from state to state for any function which does not 
vary uniformly. 

In general, therefore, it is a function of the variable. 



DIFFERENTIAL AND DIFFERENTIAL COEFFICIENT. 



57 



It may have values from — x to -}-x . 

Having represented a function by the ordinate of a curve, the 
differential coefficient is equal to the tangent of the angle made 
.with the axis of abscissas, by a tangent to the curve at the point 
corresponding to the state considered, £ 55. 
Thus, assuming PJ?= 
the differential co- 
efficient of the function, 
represented by the ordi- 
nate of the curve AM, 
at the state PA, is equal 

to 

dy 

tenX£A=tanQA£= -jg 

It should be noticed 




that '-f- is independent of the value 
assumed for the differential of the variable; for if PR'=dx, then 
Q'D=dy i and we have, as before, '-f- =tanXJEA. 

In this illustration the function is an increasing one, and its 
differential coefficient is positive, since it is equal to the tangent of 
an acute angle. 

In case the function represented 
by the ordinate of AM, is a decreas- 
ing one, its differential coefficient 
corresponding to PA is negative, 
since the anele XEA is then obtuse. 



\A 


Q 


NsJE 






V 


dy 

B 

R ^ 




\m 

P &x x 


X 



67. The following facts should now be apparent concerning a 
differential of a function. 

It is zero for a constant. 

It is constant for any function which varies uniformly. 

It is a function of the variable for any function which does not 
vary uniformly. 

Its value depends upon that of the differential coefficient, and 
that assumed for the differential of the variable. 

It may have values from — x to -j-x . 



58 DIFFERENTIAL AND DIFFERENTIAL COEFFICIENT. 

It will be numerically greater or less than the differential 
coefficient depending upon whether the differential of the variable 
is assumed greater or less than unity. 

It has the same sign as its differential coefficient. 

68. If the differential of the variable is assumed equal to. the 
unit of the variable, the differential of a function and the corres- 
ponding differential coefficient, will have the same numerical value. 

Thus, if -j- =z2, and ^=iinch, we have, ~dx=2 inches. In 

such cases the differential of the function expresses the rate in 
terms of the unit of the variable; and since it is more definite, it is 
frequently used instead of the differential coefficient. 

To illustrate, let s denote any 
g variable distance regarded as a func- 

tion of time, giving s=f(t). Assum- 
ing any convenient length to represent 
the unit of t, we may, by substituting 
s for y and / for x, § 20,- determine 
a line, as AM, whose ordinate repre- 
sents the given function. 

If PR=dt represents one hour, 

ds 

— dt=Q£ represents the change that 

s would undergo in one hour, from the state represented by PA, 
were it to retain its rate at that state; and is more definite than 

ds 

the corresponding abstract value of — . 



69. The differential coefficient of the variable distance from 
any origin to a point in motion, regarded as a function of the time 
of the motion, is called the velocity of the moving point with 
respect to that origin. 

Representing the variable distance by s, the symbol for the 

velocity is — . 

For the reasons described, velocity is measured by the product of 
— and the distance assumed to represent the unit of time. 



A 


// M 


ds 

Q 

R T 





S 

P dt 




VELOCITY AND ACCELERATION. 59 

That is, the measure of the velocity of a point in motion at any 
instant, in any required direction, is the distance in that direction, that 
the point would go in the next unit of time, were it to retain its rate at 
that instant. 

It should be noticed that the 
distance referred to above, and repre- 
sented by s, may, or may not, be esti- 
mated along the line or path upon 
which the body moves. Thus, if a 
point moves from A towards B, and 
the velocity at any point, as C, in the 
direction AB is required, the distance 

s is estimated along the path described; but if the rate or velocity 
with which a point, moving from A to B, is approaching D is 
required, s must represent the variable distance from the moving 

ds 

point to D, in order that — - shall be the required rate of motion. 

Since velocity is a rate, it is constant in uniform motion, and a 
variable function of time in varied motion. § 56. 

The differential coefficient of velocity regarded as a function of 

time is called acceleration. It is denoted by — , in which, v repre- 
sents velocity. 

Acceleration is generally expressed in terms of the distance 
which represents the unit of time. 

The differential coefficient of any varying angle regarded as a 
function of the time is called angular velocity. 

Representing any varying angle by 6, and its angular velocity 

by 00. we rtave &?= — . 

■* ' at 

If s denotes the varying arc, of a circle whose radius is r, which 
subtends 0, we have 

5 db _lds* 

0=— • hence, w= -r — -37 . 
r 3 at rat 

That is, angular velocity is equal to the actual velocity of a point, 
describing any convenient circle about the vertex of the angle as a 
centre, divided by its radius. 

*Assume this result for the present. 



6o 



PRINCIPLES OF DIFFERENTIAL COEFFICIENT. 



It is customary in applied mathematics to consider the radius 
equal to the unit of distance used in any particular case. Angular 
velocity will then be measured by the actual velocity of a point at 
the unit's distance from the vertex. 

The differential coefficient of angular velocity regarded as a 
function of time is called angular acceleration. 

It is denoted by — — , in which go represents angular velocity. 



70. Let y—PA represent any state of any increasing function 
of x; and y' the new state corresponding to an increment PP f =h 




of the variable. 



will be positive, provided h is assumed suf- 



ficiently small, and will remain so as h approaches zero, § 14. 



Hence, § 31, Theorem I. 



limit [ >'— y ~\ _ dy_ 



tan XPA, is 



positive. 

That is, the differential coefficient corresponding to a?iy state of an 
increasing function is positive. 

Let y=P x A t represent any state of a decreasing function; and 
y' its new state due to an increment of the variable equal to P 1 P 1 f =h. 

Then ^-7-^ will be negative, if h is small enough, and will remain 
so as h approaches zero. 

Hence, «J* p=2] = g = tan XE X A X , is negative. 

That is, the differential coefficient corresponding to any state of a 
decreasing function is negative. 

It follows, that a function is increasing when its differential 
coefficient is positive, and decreasing when it is negative. 



PRINCIPLES OF DIFFERENTIALS. 



61 



If for any value of the variable 
the differential coefficient is zero, 
the function is neither increasing 
nor decreasing, and the tangents at 
the corresponding points of the line 
whose ordinate represents the func- 
tion, are parallel to the axis of X. 

If the differential coefficient is infinity, the rate of the function is 
infinite; and the tangents at the corresponding points of the line 
whose ordinate represents the function, are perpendicular to the axis 
of X. 




71. Let cp (r) and ip (x) represent any two functions of the 
same variable which are equal in all their successive states, giving 
cp [x)--ip (x) .... (i). Increase x by Ax, and we have 
<p (X+ Ax)=ip (x-\- Ax) .... (2). 

Subtract (1) from . (2), member from member; divide both 
members of the resulting equation by Ax, and we have 

<&+*'>-&) = ^+AX)-» W for aU va[ues Qf and ^ 



Hence, 



limit 



r <p{x+ Ax)— (p(x) l 
AxB^O L Ax J ~ 



limit 

Ax«H 



o[ 



j){x+ Ax)—lp{x) 
Ax 



y 



§ 31, Theorem II. Therefore, 



dcp(x) diP{x) 



also dcp{x) = dip (x) 



dx dx 

That is, if two functio7is of the same variable are equal in all their 
successive states, their corresponding differentials are equal. 

Cor. If any two corresponding states of two differentials of func- 
tions of the same variable, are unequal, the functions are not equal in 
all their successive states. 



72. Having given f(x)±C, in which C represents any con- 
stant, we have, by the application of the general rule § 53, 



limit 



i[ 



[f(x+A)±C]-[f(x)±C]-\ _ Umit V f{x+h)-f(x) 



Hence, 



d(f{x)±C) 
dx 



df{x) 
dx 



- kw^>0 L h J 

and d(f(x)±C)=df(x). 



62 



PRINCIPLES OF DIFFERENTIAL COEFFICIENTS. 



That is, the differential of a function plus or minus a constant is 
equal to the differential of the function. 

Cor. If two corresponding differentials are equal, it does not follow 
that the functions from which they were derived are equal. 



and 



F{y) 



(0. 



73. Let y=/(.x) . . . (i), 
be direct and inverse functions. 

In (2) increase any value of y by k, and denote the correspond- 
ing increment x' — x, of x by //. 

In (1) give the increment // to that value of x, which in (2) 
corresponded to the value of y that was increased by k, then y in 
(1) will receive an increment y'—y equal to, and corresponding to 
k, the assumed increment of y in (2). § 46. 



Hence, 



1 

x'—x 



Taking their limits under the law h^^O, which requires /&*»-K), 
we have 7 limit A R^z 2 ! = f^t— n 

hw>-*0 L h J limit x'—x l- 
kv*, \Q L Jr. J 

Hence, 



dy 
dx 



dx 

dy 



That is, corresponding to any value of x, the differential coeffi- 
cient of y regarded as a function of x, is the reciprocal of the differ- 
ential coefficient of the inverse function. 

To illustrate, let the function y 
be represented by the ordinate of the 
curve AM. Assume dx=PR, and 
from the figure we have, correspond- 
ing to the state PA, 




QB 
PR 



dy 

-r =tan QAB. 



The inverse function will be re- 
presented by the abscissa of the curve 
AM regarded as a function of the 
ordinate, and assuming dy—KL, we have for the state KA, corre- 
sponding to A, 



HE dx 
AH = dy= tanEAH ' 



EAH=W°—QAB. 



PRINCIPLES OF DIFFERENTIAL COEFFICIENTS. 63 

Hence, 

X*nQAB'=cotEAH=^ I]l ; or £= ^- 

dy 

It should be observed that, in general, dy in the first member of 
the above equation is not the same as dy in the second; for the 
first is the differential of y as a function, which, in general, is a 
variable; and the second is a differential of y as the independent 
variable. The same remarks apply to dx, in the two members, 
taken in reverse order. 

The figure illustrates the differences referred to. 

74. Let y be an implicit function of x, the relation being 
given by the two equations 
y=/(u) .... (1). u=<p(x) .... (2). 

Increase any value of x by h, and denote the corresponding 
increment u' — //, of u by k. 

In (1), increase by k that value of u which in (2) corresponds 
to the value of -x that was increased by h, then y in (1) will 
receive its corresponding increment y' — y. 

Hence, since u' — u=k, and the increments of x, u, and y 
correspond to the same value of x, we have 

y' — y_y' — y **■' — u 
h = k X li * 

Taking their limits, under the law //^^0, which requires /£«*-H), 
we have "?» P-^l = ] imi ' P^l x J™' \^-\ 

tt dy dy du 

Hence, -f- — -~ X -j- . 

dx du dx 

That is, corresponding to any value of x, the differential coefficient 
of y regarded as a function of x, is equal to the product of the differ- 
ential coefficient of y regarded as a function of u, by the differential 
coefficient of u regarded as a function of x. 

Similarly, having y = f(u), ti = (p(x), x=ip(s), we find 
dy dy du dx 
di du dx ds J 

and the same form holds true whatever be the number of the inter- 
mediate functions. 



64 PRINCIPLES OF DIFFERENTIAL COEFFICIENTS. 

If we have y—f{u) . . . (i), and x = ip(u) . . . (2); 
(2) may be written u = cp(x) . . . (3). 

Hence, from (1) and (3), y=^Xj. 

7 w/ dx du ax 

But '§73, g = |- Hence, f = |_. 

That is, corresponding to any value of x, the differential coefficient 
of y regarded as a function of x, is equal to the quotient of the differ- 
ejitial coefficient of y regarded as a function of 21, by the differential 
coefficient of x regarded as a function of u. 



Examples. 
Given 

or dv <• 

1. y=au d , u—ox. ........<*.— — 2ab 2 x 

dx 

2. z=ay 2 , y % = 2px. . •— =2 ap. 

dy_ 

3. y=f(u), *=,*«), »=*,) £ = -| X |- 

2 2 <^J' J^J 2 

4. y—u*, X=3U, X—2s a . ..... — = • 

<& 9 

eta 

5 . „=/(«), «=^), *=*,) | = |x'f : 

dy dj; 

6. y=f{ti), v=cp(u), v=ip(s), z=F(s), z=F 2 (x). ^ = W X "^ X ^' 

dw ds 



CHAPTER V. 



DIFFERENTIATION OF FUNCTIONS OF A SINGLE 
VARIABLE. 

75. The differential of any function of a single variable may 
always be determined by applying the general rule, § 53, and 
multiplying the result by the differential of the variable. 

By applying the general rule, § 53, to a general representative 
of any particular kind of function*, there will result a particular 
form, or rule, for differentiating such functions, which is generally 
used in practice. 

76. Differential of the Product of a Function and a Constant. 

Let Cf(x) represent the product of any function by any constant 
denoted by C. 

Applying the general rule, § 53, we have 



limit r Cf(x+/i)-Cf(x) l limit r /(x+/z)-/(-r) -| _ df{x\ 

/z>^->0 L h J - W*s»->0 L h J ~ L dx 



Hence, ^. = ^; and dCf(x) = Cdf(x). 

That is, the differential of the product of a function and a constant 
is equal to the product of the constant and the differential of the 
function. 

Cor. The differential of the quotient of a function by a constant 
is equal to the quotient of the differential of the function by the constant. 

77. Differential of the Sum or Difference of any Number of 
Functions., Let y = u±s±t±etc, in which, u, s, t, etc. are any 
functions of any variable, as x. 

*Functions of a single variable only are considered in this chapter. 



66 DIFFERENTIATION. 

Applying the general rule, § 53, we have, § 31, Theorem IV, 
t ry'—yl = limit V u'—u ^ Kmit [s'—s-] limit [t—t'~\ 

L h J /i»^>0 L h J X As»->0 L A J /fe^O L A J " ! ' 

Hence 



limit 



JL— -Jt ± _i ± __ ±etc, and dy=du±ds±dt±etc. 

dx dx ax ax 



That is, M<? differential of the Sum or difference of any number of 
functions of the same variable is equal to the sum or difference of their 
differentials. 

To illustrate, let the side PM—x, of 
the rect. MN be variable, and the side 
MO=a-\-b, constant. The rect. MN will 
be equal to the sum of the two rects., MS 
and OS, giving, rect. MN—ax-\-bx. In- 
crease x by PR=dx; then from the defini- 
tion of a differential, § 63, we have 
d MN—PQy^adx+bdx. 



N 



M 



dx 



d(x' a — 2jt)=2(jt — l)dx. 



Examples. 

d(x % — 3ax+cx 2 )=(2x — $a + 2cx)dx. 
d ]^ a +x)—{x— b)+cx*— jj = (2CX— ^ &. 



78. Differential of the Product of any Number of Functions. 

Let yz be the product of any two functions of any variable, as x. 
Applying the general rule, § 53, we have 

limit V y'z'—yz l _ limit JT zAy+yAz+AyAz l 
Ax»»-»0 L Ax J Axb->0 L A^ J 

_ limit r &y ^, , A x Az l, 
31, Theorems IV and V, 



Hence 



limit r y'z'— yz-} _ H m it f Ayl 
AjB->0 L Ax J Z AxB->0 LaxJ 



AJ 



limit r , „ i limit 



Ax: 



r-i- 

L AxJ 



Therefore, 



dx 



z+ji; and ^=^-h>'> 



That is, the differential of the product of any two functions of the 
same variable is equal to the sum of the products of each function and 
the differential of the other. 



dz 
V 




/ M 


2 

s 



FUNCTIONS OF A SINGLE VARIABLE. 67 

To illustrate geometrically, let 
ONPM be a state of a rectangle with a 
variable diagonal represented by x. Two 
adjacent sides, denoted by z and y re- 
spectively, will be functions of x, and yz 
will be the variable area of the rectangle. 
Assume dx=PP, and complete the rects. 
PT—ydz, and PS—zdy. Then, since 
dyz=ydz-\-zdy, we have d (rect. OP) = rect. PT+ rect. PS. 

A consideration of the figure and the law of change shows that 
the sum of the two rects. PT and PS is the amount of change 
required by the definition of a differential. § 63. 

Let vsu be the product of any three functions of the same 
variable. Place vs=r, giving vsu — ru. 

Differentiating, we have dvsu=dru = rdu-\-udr, in which, 
dr=vds-\-sdv. Hence, by substitution, 

dvsu — vsdu-\-vuds-\-sudv ... (1) 

Having the product of four functions of the same variable, 
vsuw, we may place vsu = r, and in a manner similar to above, 
deduce 

dvsu7U—vsudw-{-vs7vdu-\-vuzvds-{-suwdv . . . (2) 

In the same way, it may be shown that the differential of the 
product of any number of functions of the same variable is equal to the 
sunt of the products of the differential of each function and all the 
others. 

79. Dividing each member of Eq. (2) by vsuw, we have 

dvsnw dtu du ds dv 

= h — H + — ' 

vsuw w u s V 

Similarly, it may be shown that the differential of the product of 
any number of functions of the same variable, divided by their product, 
is equal to the su?n of the quotie?its of the differential of each function 
by the function itself. 

80. Differential of a Fraction. Let — be any fraction, in 
which u and s are functions of the same variable. 



68 DIFFERENTIATION. 

Place — = y, then u=sy; and, § 78, du=yds-\-sdy, 
Hence, sdy=du — yds, or sdy=.du ds= sa 



s s 



r^, r j jti sdu — uds 

Therefore, dy=d - = — . 

s s* 

Hence, the differential of a fraction is equal to the denominator 
into the differential of the numerator, minus the numerator i?tto the 
differential of the denominator, divided by the square of the denomi- 
nator. 

If the numerator is a constant denoted by C, we ha've 

C__ Cds t 

s ~ s 2 

If the denominator is a constant denoted by C, we have 

u du 

d ~c = a' 

81. Differential of y m . Let y represent any function of any 
variable, and m any constant. 

i°. If m is entire and positive, y m =yyy . . , j and § 78, 

dy m —y m - 1 dy-\-y m - 1 dy -J- etc. = ?ny™- x dy. 

2 . If m is a positive fraction, equal to - j p and q being entire 
and positive, we have 

y m =y\ and (y m ) =y . Hence, i°. 

d(f D )=.q(f°) * d(f°)=py * dy. 
Whence, 

d { y^)=PJ^ = tJl^dy=lJ^dy= P -J~\y= m ;" 1 dy. 

3 . If m is negative, represent it by —n, n being entire or 

1 



fractional; then y m = — , and §80 



r- 



dy n ny n ~ 1 dy 

d()' m )=-y^ =— y,n =-ny-»-*dy=my™-idy. . . . (1). 



FUNCTIONS OF A SINGLE VARIABLE. 69 

Hence, the differential of any power of any function with a con- 
stant exponent is equal to the product of the exponent of the power, the 
function with its exponent diminished by unity, and the differential of 
the function. 

82. Substituting - for m in Eq. (i), we have 
I 1 1-j 1 — dy dy 

<r = n r Jy= - r. " <*= ~s = ;^F- 

ny n v y 

Hence, the differential of the 71 th root of any function is equal to 
the differential of the function divided by n times the 71 th root of the 
n—i power of the function. 

If;/ = 2 ; rfA /37=^_ 

2 V y 

3VJ' 2 



Examples. 

1. d (2x) 2 = 2 (2x) d(2x)=Sxdx. 

2. d(2X 2 )=2 (2X 2 ) d(2X 2 ) = l6x 3 dx. 

3 . d 4X* = 4 X 4x 3 dx = 1 6x 3 dx. 

4. d x n =nx a ~ 1 dx. 

5 . d (axf= 3 (axfd (ax) = 3 a 3 x 2 dx. 

6. d ( 3 *)-2=(— 2) ( 3 x)-*d ( 3 x)=-6 ( 3 x)~ 3 dx. 

7 . d x~ n = — n x-*- x dx. 

8. dx*=- x~^dx = dx 



2 



2 yyfx 



1 I ±_, dx 

9. dx* — -*' dx-= .r—f- 

n n \/x° l 

IO. dx~*=—±x~*dx = ^%. 

2 2 <\/x 2 



71 



y/x n ^ ' 



limit 



70 DIFFERENTIATION. 

83. Differential of log y. Let y be any function of any 
variable. Increase the variable by h, and denote, by k the corres- 
ponding increment of y. 

Applying the general rule, § 53, we have, since h and k vanish 
together, 

r log(?+*Mog.r -l limit [ WP^] ] = limit r i0g ( I+ |) ] ; 
L k J km->0 L k J &m->0 L k J 

which, placing £=y*i> equals Iimit n r log(l+;;? )l = 
Hence, — ~^ = — ,• and d log y=M — ^- . 

^7 y y 

That is, //z<? differential of the logarithm of any function is equal 
to the modulus of the system into the differential of the function divided 
by the function. 

In the Napierian system, M=i, and d log e y— — . 

84. Differential of a x . Let a be any constant, and x any 
variable. Increasing x by h, and applying the general rule, § 53, 
we have 

limit r ^+ h -^ l _ limit r ^-il _ 

h-w^oi. h J ^^^oL h \~ 

Hence, -^- =a x log e a; and da x =a x log e a dx. 

dx 

That is, M* differential of any exponential function with a constant 
base is equal to the product of the function , the Napierian logarithm of 
the base, and the differential of the exponent. 

If a=e, the base of the Napierian system, we have de x =e x dx. 

85. Differential of y z , in which y and z are functions of the 
same variable. 

Let u—y 7 -; then log e u=z log e y; and, § 78, 

du dy , , 7 

- =*j + log e ydz. 



TRIGONOMETRIC FUNCTIONS. 7 I 

Hence, du = dy z =zy z ~ 1 dy-\-y z log e _>' dz, which is the sum of the 
differentials obtained by applying; first, the rule in § 81; then, that 
in § 84. 

86. Logarithmic Differentiation. The differentiation of an 
exponential function, or one involving a product or quotient, is 
frequently simplified by first taking the Napierian logarithm of the 
function as above. 

EXAMPLES. 
I . u=x x . dti=x* (i-f- loge x) dx. 



dtc—x** x* \og e x(log e x+i)+ ~ dx. 



11= 



a/ i-\-x dx 

du— —- 



/sj\ — x (i — x)<\/l X 2 

i_ l~2x 

4. M=X*. du=X * (i log e x)dx. 

5. u = W-i) 5 _ l oge «=| loge (x-i)-f log e (x-2)-l loge Cr-3), 

\/(x-2)* \/(x- 3 y 

du _s dx 3 dx 7 _dx yx 2 -{- 30X— 97 dx 



u 2 x — I 4 x — 2 3 x— 3 12 (x — 1) (x — 2) (x — 3) 

du= 



(x— 1)^(7^2 + 30-^— 97) dx 

7 10. 

12 (x — 2) ¥ (x — 3) 3 



87. d sin <p= cos <p d <p. For, if cp be increased by A cp, we 
shall have, § 53, § 34, 

d sin cp _ limit [~ sin (<p+ A <p) — sin cp ~\ _ ximit 2 sin —£- cos y <?+ -g-^/ _ 
dcp ~ A <p»»-»0 L A <p J _ A <p^^0 1^ A cp 



limit 



. Acp 
sm ^- cos ( <p+ 



Acp 



cos 9?. 



In a similar manner, by applying the general rule, § 53, the 
differential of any trigonometric function may be determined; but 
it is perhaps simpler to make use of the relations existing between 
the functions. 

d cos cp = — sin cpd cp. For, d cos cp = d sin ( ^ — cp ) = 

= cos y— — <pjdy——<Pj = — sin cp d cp. 



72 



DIFFERENTIATION. 



d cp 
dtan(7?= CQS a = sec 2 cpdcp=(i+ tan 2 cp)dcp. For, d tan cp=d 

_ cos q)d sin <£> — sin <pd cos cp _ (cos 2 cp + sin 2 ) </<p a 7 ^ 

cos 2 cp cos 3 c> ~~ cos 2 o 



sm <y> 
cos <p 



d cp 
d cot cp = — sin 3 a? ~ — zosec 2 cp dcp = — (i + cot 2 cp) dcp. For, dcotcp= 

d (i-9») ^ 



=</tan (*-?>) 



cos ! 



(;-*) 



sm 2 © 



sin cp dcp 



d sec cp=t an cp sec cpdcp. For, dscc cp= d— - — 

^ -r 7- 7- 7- COS <£> COS <£> 

= tan <p sec cp dcp. 
d cosec cp — — cot cp cosec cp dcp. For, d cosec cp=d sec (- — <£> ) = 
= tan ( - — cp ) sec ( - — cp J d I - — cp ) = — cot cp cosec cpdcp. 

d versin cp = sincpdcp. For, ^/versin cp=d(i — cos<p)=sin cpdcp. 

d coversin<p = — cos cpdcp. For, <^covers<p^^vers ( - — cp) = 
= sin {~ 2 —cp)d(^ — cp)= — cos cpdcp. 

In order to illustrate the formulas 
for the differentials of the sin and 
cos of any angle, let AC£=cp, be 
any given angle. Assume £CN=dcp, 
and with any radius, as CO=JR, 
describe an arc, as OMN. 

Then, 

PM CP 

~n =sin<p, -jt — cos cp, arc MN=Rdcp. 

The definition of a differential, 

§ 63, in this case, requires that the 

sin cp and cos cp, retaining their rates at the states corresponding 

to cp=ACB, shall continue to change from those states while cp 

increases by the angle P?CJV=d<p. 

Draw the tangent line to the arc at Mj and lay off MT equal 
to the arc MN—Rdcp. Through T draw TQ parallel to MP, 
and through M, draw MQ parallel to OC. 




FUNCTIONS OF A SINGLE VARIABLE. 



73 



Then, QT and — MQ are, respectively, the changes that the 

lines PM and CP would undergo were they to continue to change, 

with the rates they have when cp=ACB, while cp increases by dcp. 

OT MO 
Hence, — - , and ~ are, respectively, the changes that the 

A R 

sin and cos of cp would undergo under the same requirements. 
The angle MTQ=cp. Hence, 



QT = MT cos cp — R cos cpdcp, and 



MQ = M T sin cp = R sin cp d cp, and 



QT 

-7T = ^sin <p = cos cpd q>. 

MQ 
o = dcos q> = — sin cp d cp. 



In a similar manner let the student illustrate the formulas for 
the differentials of the other trigonometric functions. 



B/ 



88. Regarding the right lines 
PM, CP, OE, O'B, etc., as functions 
of the variable angle cp, we have 







E 


/V p 


\M 


A 



dRAf=dR sin (p—R cos <pd(p. 
Rdcp 



dOE-dR tan <p-- 



cos-cp 



d CP=dR cos cp= — R sin 

Rdcp 
dO'B=dRcotcp-- 



d CE=d R sec cp—R tan cp sec cpdcp. 
J OP=dR vers q>— R sin cpdcp. 



SWi'Cp 

d CB=Rd cosec cp= — A 3 cot <pcosec cpdcp. 
dO' Q=dR covers cp= — R cos cpdcp. 

It is important to notice the difference between the differentials 
of the above lines, which depend upon the radius of the circle used, 
and the differentials of the trigonometric functions which do not 
depend upon any radius or circle. 



i. j>'=log e (sin cp). 



dy= 



Examples. 

d sin cp cos cpdcp 



sin cp sin cp 

2. y— loge 's/a 2 — x 8 . 

3. y=e nx . e = base Napierian system 



:cot (p dcp. 



xdx 



a- — x- 
dy=ne nx dx. 



74 



EXAMPLES. 



4. J— log e tan -%-. dy= 



dcp 
d tan^2- 2 cos 2 ^?- 



tan£ 



tan %- 



dcp 
sin ^» 



5. Assuming dcp— 



d sin «p: 



Vs 



we have, corresponding to cp= t 
it 



d cot cp= — 7t. 



6. Corresponding to 9)= 



d sin ^> 
dcp 

dcot cp 
dcp 



1 

~ w 

= —4- 



dcos cp= — 

Tt 

dsec cp- 7 . 

n 

- , we have 

4 

d cos cp__ 1 

d sec <p ,— 



a?tan <p= 



^ cosec <p 



VF 



*/tan 9? 



dcp 
d cosec <p 



=4- 



=— V2- 



z/ = 


sin m x 
cos"*' 


dfo 


.T. 


COS.T 



dcp dcp 

r sin x ~| 
du=x smx I cos* loge* + <&. 

. '. log e w=//z log 6 sin * — n log e cos */ and 
m m ~ 1 x «sin in + 1 jr" 



-1 



9. 


y — sin^, 


10. 


jj/=sin 3 <p, 


II. 


J=COS 2 <£>, 


12. 


_j/= cos 3 <p, 


13. 


7= tan 2 <p, 


14. 


j= tan 3 cp, 


15. 


y — COt* cp, 


16. 


jy^cot 3 ^, 


17. 


y '■= sec^cp, 


18. 


y = sec 3 cp, 


I 9 . 


y = cosec s (p, 


20. 


y= cosec 3 cp, 


21. 


y = versin 2 <p, 


22. 


y = versin 3 £>, 


23. 


y = covers 2 cp, 


24. 


y = covers 3 cp, 



7 . 7 I lit Sill 

dx . . du= 

|_ cos n ~ x x ' cos n_ 

dy = 2 sin cp cos cpdcp. 
dy = 3 sin 2 cp cos cpdcp. 
dy = — 2 cos cp sin cpdcp. 
dy = — 3 cos 2 <psin cpdcp. 
dcp 






^j/ = 2 tan <p 
d£j/ = 3 tan 3 <p 



dcp 
cos 2 <p ' 
dcp 



sm'cp 
dcp 



dy = — 2 cot 93 

dy = — 3 cot 2 <z> 

dy = 2 sec cp tan cp sec cpdcp: 

dy= 3 sec 2 <p tan 93 sec cpdcp. 

dy = — 2 cosec 9) cot cp cosec cpdcp. 

dy = — 3 cosec 8 <p cot (p cosec cpdcp 

dy = 2 versin 9) sin cpdcp. 

dy = 3 versin 2 cp sin cpdcp. 

dy = — 2 covers 95 cos cpdcp. 

dy = — 3 covers 2 ^? cos cpdcp. 



-# 



INVERSE TRIGONOMETRIC FUNCTIONS. 75 

du , 

RQ i°. dsin _1 u= — . . Let <p=s\n-' i zi; then tt — sm<p, and 

".**• V 1 — 113 

tf« ^(p I \ 

-j— = cos cp. Hence, S 73, —7- — = , / r-^— = 

dcp ^ '■ '•*" du cos cp ±yi — sin 2 <p 

1 du 

■= . > or ^/<p =:^sin _1 ?^= "^jj — , 

±yi — z* 2 ^V 1 — ^ : 

— du /7T \ — </« 

2°. dcos~ 1 u= — , For dcos~ 1 u=d[ - — sin -1 ?* I — . -> 

\/l— U 2 V2 / ^/i— U* 

du 
3 . d tan _1 u= TZTTTT • Let q)=ta.n~ 1 u; then z*=tan<£>, and 

du 1 <f<p I 

~T~ — iTT" Hence, S 73, "-7- = cos 2 <p = r— = 

aT<p cos^<£> ' ° /J ' ^ * sec^tp 

1 1 d^ 

= — — — 5 — — T , o , or dcp=dta.n~ x u = — ; — :> . 
i+tan-<p i + u~ ' ^ i + z^ 

— du /7zr \ — du 

4°. d cot^u^ I+lia . For, ^/cor 1 «=*? ^- — tan" 1 ^ = ^-p^- 2 * 

du 

5 . d sec _1 u= . ■ Let <p=sec x u; then u=seccp, and 

uyu 2 — 1 
<fo ^ 1 

3 — =sec<p tan<z>. Hence, s> 73, -7— — 7 — 

</<p v v- > s /J' ^ sec(ptan<p 

1 ; , , , ^ 

. ~= , ', or dcp=zdsec~ 1 u= 

sec^Y 5602 *? 5 — x ^v^ S — I u^/u' 4 — 1 

— du /7T \ 

6°. dcosec _1 u= , For dcosec~ x u=d ( - — sec~ 1 z^) = 

u/y/u 2 — I \2 / 

— du 



u<yzc~ — I 

du 
7 . d versin~ 1 u= ~~~7 2 • Let <p=versin x u; then u— versing, 

<y/ 2U — U" 

dfo </<p I I x 

and -j— = sin <p. Hence, § 73, 



dcp r\ ,v,so, du sin(p v / I _ cosa ^ 



/y/i — (1 — versep) 8 ^J 2 vers cp — vers 2 <p ^2u — u 2, * 

du 

or dcp=dvers~ 1 u= <— , =-■ 

y2« — w 3 

*The sign depends upon that of cos cp. The formula is generally written 
with the plus sign only, which corresponds to angles ending in the I st or 4 th 
quadrants. 

Formulas 2°, 5 , 6°, 7 and 8°, also involve the double sign, but are generally 
written as indicated above. 2°, 5 and 7 , as given, correspond to angles ending in 
the i 3t or 2 nd quadrants; and 6° and 8° correspond to angles ending in the I st or 
4 th quadrants. 



7 6 



DIFFERENTIATION. 

— du 



— du /it \ 

d covers 1 u = — . For ^covers -1 u=d I — — vers -1 ?* ) = 

V2U-U 2 \2 / 



-du 



V* 




90. Regarding cp as a function 
of the line .PJ/, denoted by y, we 

have ^=sin -1 ^-. Hence, 



d <p: 



R 



dy 



Similarly, having CP=y 

Similarly, having OE=y 

Similarly, having 0'B=y 

Similarly, having CE=y, 

Similarly, having CB=y, 

Similarly, having PO — y .'. <p = versin 

Similarly, having 0'Q=y, 



— dy 



cp = 'co$, — , we have d<p= 



yR*—y % 



-1 y Rdy 

<p=tan -77, we have dcp— ^ a _j_ 2 



<p=cot ^-, we have d<p= 

m=sec J- , we have dcp— 
R ^ 



-Rdy 



R*+y> 



Rdv 



y<s/y*—R* 

-l v — Rdy 

<z>=cosec — , we have dcp= -, 

R ^ yyyX—R 2 



R 



we have dcp = 



dy 



<Z) = coversin — , we have dcp — 
R 



\/2Ry—y' i 
-dv 



\ / 2Ry—y' i 



Y 




4- 1 


4' 


d# 


M 








R' 
X 


f 


P 


Ax P 


R 



9 1 . Differential of an Arc of a Plane 

Curve. Let s represent the length of a 
varying portion of any plane curve in the 
plane XY. It will be a function of one 
independent variable only, § 19, which we 
may take to be x. 

Assume any point of the curve, as M, 
and increase the corresponding value of 
x=OP, by PjP'ax. As=MM' will be 
the corresponding increment of s. 



ARC OF A PLANE CURVE. 77 

Then, § 37, 

d ± __ limit [AJl _ 1 . 

dx Ax^-^Ola^J cos R' M T 

dr 

Assume PR=dx. Then, § 63, R'T—dy, and cos R'MT= -^ 
Substituting in above, 

ds MT 



dx dx 



MT=ds=/v/dx 2 + dy 3 



The double sign is omitted because s may always be considered as an 
increasing function of x. 

That is, the differential of an arc of a plane curve is equal to the 
square root of the sum of the squares of the differentials of the coordi- 
nates of its extreme point. 

If s were to change from its state corresponding to any point, 
as- M, with its rate at that state unchanged, the generatrix would 
move upon the tangent line at Mj hence, MT= ^/dx 2 +dy 2 
represents ds in direction and measure. 

In order to express ds in 'terms of x and dx only, substitute 
for dy its expression in terms of x and dx determined from the 
equation of the curve. 

Similarly, ds may be expressed in terms of y and dy only. 
Thus, let s be an arc of the circle whose equation is x 2 -\-y 2 =4. 
Solving with respect to y, and differentiating, we have 

xdx 

dy= T 



V4— x 



•x< 



tt j A / x 2 dx* 2dx A ds 2 

Hence. d^\/ dt , + —— = ^= t and _ = ^= 

(t) — (7) =- (t) =-■ 

\dx/cc=—2 \dx/x=0 \dx/x=2 



* The square of the differential of a variable represented by a single letter, is 
generally written as indicated in the above formula; and is similar in form to the 
symbol for the differential of the square of the variable. 

Similarly, the n th power of the dx is generally written dx n . 

A knowledge of the formula, and of the associated symbols, removes the 
ambiguity in such cases. 



78 



DIFFERENTIATION. 



That is, at the points where the circum. cuts X the rate of s 
with respect to x is infinity; while at the points where it intersects 
Y its rate is the same as that of x. 




92. Differential of 
any Arc. Let s represent 
the length of a varying 
portion of any curve in 
space. It will be a func- 
tion of one independent 
variable only, § 19, which 
we may assume to be x. 
Through any assumed 
point of the curve, as M, 
draw the ordinate MJV; 
and through JV, the point 
where it pierces XY, draw NP parallel to Y. OP will be the 
value of x corresponding to M. Increase x=OP by PP'=Ax, 
and through P' pass a plane parallel to YZ, intersecting the given 
curve at M' . As— arc MM' will be the increment of s correspond- 
ing to the assumed increment of x. 

Draw the chord MM' and the ordinate M'K. Through M 
draw MQ' parallel to a right line drawn through N and K; and 
through N draw NN' parallel to X. Then, N'K=&y, and 
Q'M'= Az, will be the increments of y and z corresponding to 
Ax; and we have chord MM'=^/(&x)*+(Ay)* + (Az)*. 
Hence, § 31, Theorem X. 



ds_ 

dx 



limit 

Axn-> 



it f arcMM '-l ^ Hl 
>->0 L Ax J Ax 



lj ni il rchor&MM' 
■0 L Ax 



_ limit V V(Ax) s + (Ayy 



■{AzY 



] 



limit 
A x »»-»i 



o^+m°+m 



\f I+ f d lY + ^V; hence, ds^V^x^dy^dz 2 . 



PLANE AREA. 



79 



In the above deduction s may be a curve of single, or of double 
curvature. The increment As may, or may not, lie in the project- 
ing plane of the chord MM' . 

If not, the projection of the chord MM' on the plane XY will 
change direction as Ax approaches zero, but the above relations 
will not be affected thereby. 



93. Differential of a Plane Area. 

Let u represent the area of the plane 
surface included between any varying 
portion of any plane curve, as AM, the 
ordinates of its extremities, and the axis 
of X. 

Regarding u as a function of #,'§ 21; let x=QP', be increased 
by P'P"= Ax. P'MM"P" will be the corresponding increment 
of u. Hence, § $&, 




du 
dx 



limit f P'MM"P" 



Ax 



-y- 



du=y dx*. 



That is, the differential of a plane area is equal to the ordinate of 
the extreme point of the bounding curve into the differential of the 
abscissa. 

To illustrate, let u represent the area 
BAMP, and PR—dx- then du = ydx= 
rect. PQ, which fulfils the requirements 
of the definition of a differential, § 63. 

Similarly, it may be shown that xdy is 
the differential of the plane area included 
between any arc, the abscissas of its 
extremities, and the axis of Y. 

In case the coordinate axes are inclined to each other by an 
angle 6, we have ^z/^sin 0dx, or du=x s'm 6dy. 




*It is important to notice and remember that ydx is the differential of a plane 
area bounded as described; and that it is not, in general, the differential of a plane 
area otherwise bounded. 



8o 



DIFFERENTIATION. 



In order to express du in terms of x and d x, substitute for y, 
or dy, its expression determined from the equation of the bounding 
curve. 

Thus, if a 2 y 2 -\-b 2 x 2 = a 2 o 2 is the equation of the bounding curve, 
and du. 



we have y— - ^/a a — ; 



aV"'~ 



x 2 dx. 



94. Differential of a Surface of Revo- 
lution. Let the axis of X coincide with 
the axis of revolution; and let PM=s, be 
any varying portion of the meridian curve 
in the plane XY, Through M draw the 
tangent MT, the ordinate MP, and the 
right line MR' parallel to X. Let u 
represent the surface generated by s; and 
i regarding it as a function of x, § 23, let 

x — OP be increased by PP'=Ax. 
MM'— &s, will be the corresponding increment of s; and the 
surface generated by it will be the increment of the function u 
corresponding to A^. Hence, § 39, 



^ r 


xC 


dy 




>^~H 






M 






R' 






r 


P AX P 


R 


X 



du _ u m i t T sur. gen. by arc MM' "1 
dx Axi->0 L Ax J 



1Tt\> 



cos R'MT- 



Assume PP=dxj then R'T—dy, MT—ds, and cos R'MT- 
Substituting this expression for cos R' MT in above, we have 

du 2 ity ds 



dx 



dx 



and du = 2 7ry ds=2 7ry\/dx 2 +dy 3 



Hence, the differential of a surface of revolution is equal to the 
product of the circum. of a circle perpendicular to the axis and the 
differential of the arc of the generating curve. 

Similarly, it may be shown that 27tx* s Jdx % +dy i is the differential 
of a surface of revolution generated by revolving a plane curve 
about the axis of Y. 

In order to express du in terms of a single variable and its 
differential, find expressions for y and dy in terms of x and dx, 
or of dx in terms of y and dy, from the equation of the generating 
curve; and substitute them in the formula. 



VOLUME OF REVOLUTION. 



81 



Thus, if y 2 —2px is the equation of the generating curve, we 



have 



v=vV* and dy- 



pdx 

V ' 2pX 



Hence, 



dl( 



= 2 7T^/^\/ dx2 + tl^ =2 7t {2px+f) % dx 



2pX 




95. Differential of a Volume of 
Revolution. Let the axis of X coin- 
cide with the axis of revolution; and 
let BM be any varying portion of the 
meridian curve in the plane XY. 
Through M draw the ordinate MP, 
and the right line MQ' parallel to X. 

Let v represent the volume generated by the plane surface 
included between the arc BM, the ordinates of its extremities, and 
the axis of X. Regarding v as a function of x, § 27, let x be 
increased by PP'= Ax. The volume generated by the plane 
surface PMM'P' will be the corresponding increment of the 
function v. 

Then, § 40, 



dv limit ["vol. gen. by PMM'P'~\ 



Hence, the differential of a volume of revolution is equal to the area 
of a circle perpendicular to the axis into the differential of the abscissa 
of the meridian curve. 

Similarly, it may be shown that 7rx 2 dy is the differential of a 
volume of revolution generated by revolving a plane surface about 
the axis of Y. 

In order to express dv in terms of a single variable and its 
differential, determine an expression for y in terms of x, or of dx 
in terms of y and dy, from the equation of the meridian curve; and 
substitute them in the formula. 

Thus, if ^ 2 -|->' 2 — 2^?jc=0 is the equation of the meridian curve, 
we have dv—7t (2 Px — x 2 ) dx; or since 



dx= 



ydy 



dv— 



ity z dy 

\/y 2 +x 2 



82 



DIFFERENTIATION. 



96. Differential of an Arc of 
a Plane Curve in terms of Polar 
Coordinates. Let r—f{v) be the 
polar equation of any plane curve, 
as BMM', referred to the fixed 
right line PD, and the pole P. 
Let BM—s, be any varying por- 
tion of the curve, and PM=r, 
the radius vector corresponding 
to M. Regarding s as a func- 
tion of v, §19, let v be increased 
by MP M'—Av. The arc 
MM' = as will be the corres- 
ponding increment of s. With P 
as a centre, and PM as a radius, 
describe the arc MQ f . Denote PM* by r'j then Q'M'=r'—r, 
will be the increment of r corresponding to A v. Through M 
draw the tangent MT, and the chords MM' and MQ' . 




Then, § 41, we have 



(i i_ _ limit 
dv A v ^->0 

Hence, 



MM 



_ J _ limit \/( r ' — r \ a 
J A^s^oV \ AV ) 

ds=\/dr 2 +r a dv a . 



-f-r* 



-V' 



+r' 




If the radius vector PM coincides 

with the normal to the curve at M, the 

corresponding tangent to the arc MQ' 

will coincide with MT; and § 41, 

ds _ limit r &rc MM' -l . . 

-7- — A „ =.r, gi vmg as = razz. 

dv Avm-^Q L A^ J & fe 

In this case dr=0, because the 
motion of the generatrix at the point 
considered is perpendicular to the radius 
vector. 



Since the radius of a circle is always normal to the arc, the 
differential of an arc of a circle regarded as a function of the corres- 
p07iding angle at the centre, is equal to its radius into the differential of 
the angle. 



PLANE CURVE IN POLAR COORDINATES. 



83 



Let BM=s, be an arc of a circle 
subtending the angle BCM~v. 
Assume MCQ—dv; then will the arc 
MQ—rdv. The direction of the 
motion of the generatrix at any point, 
is along the corresponding tangent 
to s; hence, by laying off from M, 
upon the tangent at that point, a 
distance equal to ds=a.vc AfQ=rdv, 
we have ds represented in measure and direction. 




In order to represent ds in the 
general case, let BM be the given 
curve, P the pole, M the assumed 
point, and MPM'— dv. If r 
were constant, as we have seen 
in the case of a circle, MT'^rdv 
would be ds- but, in general, ds 
is affected by a uniform change in 
r, in the direction BM, equal to 
dr. To determine it, we have 




dr_ 
dv 



limit 
Avm-^> 



[ Q'M' 1_ limit r Q'M' -1 



At T' draw T'T parallel to BM; then 



i—L = tan T 'MT= — • 

T'M rdv 



Hence, ~= r — — = — — , and dr—T'T. 
dv rdv dv 



Hence, M T=ds= ^/dr^+r^dv 2 , represents ds in measure and 
direction. 

In order to express ds in terms of a single variable and its 
differential, find expressions for r and dr in terms of v and dv, 
or an expression for dv in terms of r and dr, from the polar equa- 
tion of the curve: and substitute in the formula. 



8 4 



DIFFERENTIATION. 



97. Differential of a Plane 
Area in terms of Polar Coordi- 
nates. Let u represent the area- 
of a varying portion of the surface 
generated by the radius vector 
PM revolving about the pole P. 
Regarding u as a function of v, 
B § 22, let MPM f — av. The area 

MPM' , represented by A u, will be the corresponding increment 

of u. 




Hence, § 42, 



du 
dv 



limit 
Az>»»-»0 



— - = — ; and du= 

LawJ 2 



r 2 dv 
2 * 

describe the arc of a circle 

dv 



To illustrate, with PM- 
MQ-^rdv corresponding to MPQ=dv; then du—'— = area of 
the circular sector MPQ. 

du may be expressed in terms of v and dv, by substituting for 
r its value in terms of v, determined from the polar equation of the 
bounding curve. 



Problems in Rates. 



1. Having j 2 =5/ 3 , find the velocity and acceleration when t=2 seconds; 
/ = 3 seconds. 

2. Find the angles that a tangent to the curve x s =6/ 2 + 3jK+i, at the point 
(8', 3), makes with the axes X and Y, respectively. 

3. Find the rate of change of ( \/ x + — ^ ) when x = 3. 

4. Find the angles that a tangent to the curve y=logx, at the point (1, o) 
makes with the coordinate axes respectively. 

5. Find the rate of change of the ordinate of a circle with respect to the 
abscissa. 

6. Same of an ellipse. 

7. Same of a parabola. 

8. Same of an hyperbola. 

9. At what rate does the. volume of a cube change with respect to the 
length of an edge? 



EXAMPLES IN RATES. 



85 



10. Find the rate of change of a logarithm in the common system when the 
number is 12. 

11. Same for the numbers ^, ^, 157, 3227. 

12. The area of a circle is increasing 5 sq. ft. a second; find the rate per 
second of its radius when the radius is 3 feet. 

12. The side of a square is increasing 3 in. a minute; find the rate per 
minute of its area. 

14. The relation between the time denoted by t ; and the distance, repre- 
sented by s, through which a body, starting from rest, falls in a vacuum near the 
earth's surface, is expressed, very nearly, by the equation s=i6.i t~; s being in 
feet and t in seconds. Construct a table giving, the entire distance fallen through 
in 1 second; in 2 seconds; in 3 seconds; and in 4 seconds; the distance passed over 
during each of the above seconds: the velocity and acceleration at the end of each. 



Time in 
Seconds. 


Entire Distance 
in Feet. 


Distance each 
Second. 


Velocity. 


Accelera- 
tion. 


1 


16. 1 


16.I 


32.2 


32.2 


2 


64.4 


4S.3 


64.4 


32.2 


3 


144.9 


80.5 


96.6 


32.2 


4 


257.6 


112. 7 


128.8 


32.2 



The following general outline of steps may assist the student in solving 
problems involving rates. 

i°. Draw a figure representing the magnitudes and directions under consid- 
eration; and denote the variable parts by the final letters of the alphabet. 

2°. Following the word given, write, with the proper symbols, all known 
data; and after the word required indicate the symbols for the required rates. 

3 . From the relations between the magnitudes, find an expression for the 
function whose rate is required, in terms of the variable. 

4°. Differentiate and determine values or expressions for the required rates. 

In case an explicit function of a variable cannot be found, make use of the 
principles in § 74. 

15*. A man 6 feet in height, walks away from a light 10 feet above the 
ground, at the rate of 3 mi. per hour. At what rate is the end of his shadow mov- 
ing, and at what rate does his shadow increase in length ? 



* Examples 15 to 22 are from Rice and Johnson's Calculus. 



86 



EXAMPLES IN RATES. 




Let x= A M=distance from foot 
of light to man. 

Let y=AB=:dista.nce from foot 
of light to end of shadow. 

Let s=MB=- length of shadow. 



hr. 



M 




3 Let *= 


:time in hr 


Given. 

AL=ioit., MC^bit 


, Z>Z^ 4 ft., 


dx i 


dy 
Required, -77 , and 


ds 

dr 






From similar triangles, 








y: xv. 10: 4, 


•'• 


v= ~ x, and 
" 2 


dx 2 


dy dy dx 

§74, di := dx X d7 


5 w 3 mi. 1 mi. 
~ 2 X hr. ~ 7 * hr. ' 




s: x:: 6: 4, 




j= ~ x, and 
2 


afr _ 3 

dx ~ 2 ' 



*& aft *£r 3 mi. mi. 

- = -X- = ^X3^=4^ 



dt 



dx dt 



hr. 



hr. 



16. A vessel sailing south at the rate of 8 mi. per hour, is 20 mi. north of a 
vessel sailing east at the rate of 10 mi. an hour. At what rate are they separating 
at the time ? at the end of 1^ hrs. ? at the end of 2\ hrs. ? When are they neither 
separating from nor approaching each other? 

Let t — time in hours from the given epoch. 
Let AB— y=20 — 8 /—distance of I st ship from BC 
t hours after the given epoch. 




/du\ 
\dt) t=0 = 



Let BC=x=iofc 
the same time. 



distance of 2 nd ship from BA at 



Let u=AC=\/x :i +y'' s = y^oo — 320/+i64/ s . 



dy 
Given, -^ = - 

du 
Required, -77 — 



mi. 
hr7 



dx mi. 

dt hr. 



— 160+164 t 



y^oo — 320 /+164 t s 



mi. 
hrT 



'du\ I mi. 

^dt) t=\\~^ 17 hr - 



(£)<=** =°- 



* ^ hr~ mcucates 3 m i- P er nour - 



EXAMPLES IN RATES. 87 

17. The rate of increase of a side of an equilateral triangle is )/ z inch per 
second, find the rate of its altitude per second. If the rate of a side is 3 feet per 
second, find the rate per second of the area when the side is 10 ft. 

iS. A man walks on a straight line, 5 ft. per second. How fast does he 

approach a point 120 ft. from his path in a perpendicular to it, when he is 50 ft. 

from the foot of the perpendicular ? , 2 ft. 

1 jy 

sec. 

19. A ladder 25 ft. in length leans against a wall; the bottom is drawn out 
2 feet per second, at what rate is the top descending when the bottom is 7 ft. from 
the wall? in. 

sec. 

20. Two locomotives are moving along two straight railways which intersect 
at an angle of 6o°; one approaches the intersection at 25 miles per hour, and the 
other is leaving it at the rate of 30 miles an hour, find rate per hour at which they 
are separating from each other when each is 10 miles from the intersection. 

„i mi. 
hr. 

21. A street crossing is 10 ft. from a lamp situated directly over the curb- 
stone, which is 60 ft. from walls of opposite buildings. If a man walks across to 
opposite side at the rate of 4 miles per hour, at what rate per hour will his shadow 
move upon the walls when he is 5 ft. from the curbstone ? When he is 20 ft. 

from the curbstone ? , mi. A mi. 

06 6 • 

hr. hr. 

22. The radius of a sphere is decreasing 2 in. per second; find the rate of 
its surface, and volume. 

23. In the parabola y s =gx, find the rate of y with respect to x when ar=4. 
What value will x have when rate of y equals that of x? When rate of y is the 
greatest ? When the least ? 

24*. A boy is running on a horizontal plane towards the foot of a tower 
60 ft. in height. How much faster is he approaching the foot than the top of the 
tower ? How far is he from the foot when he is approaching it twice as fast as he 
is the top ? At 100 feet from foot, how much faster is he approaching it than the 
top? 

25. x 2 =2pz is the equation of a parabola OM. 
A point starting from 0, moves along the curve in such 
a manner that 2=16.1 1~; in which z is expressed in 
ft., and t in seconds. Find the rate of x with respect 
to t. 

Given, 

dx P P . dz 

— = — ;- ;:zr = — t~ — = 32.2/, 

dz V2/2 y32.2// dt 




* From Olney's Calculus. 



88 EXAMPLES IN RATES. 



dx 
Required, -,- 



dx dx dz 



ax ax az r /— 

— = — X — = / —X 32. 2 ^=^32.2/. 

dt dz dt V32.2// 2 

26. One ship was sailing south at 6 -— - , another east at 8 — l . At 4 P. M . 

hr. hr. 

the second crossed the tracks of the first at a point where the first was 2 hrs. before. 
How was the distance between the ships changing at 3 P. M. ? When was the 
distance between them not changing ? 

27. A ship is sailing south 6o° east, 8-—^ ; find the rate of her latitude and 

hr. 
longitude. 

28. A point P moves in a straight line away from a point B at the rate of 

g — - ; find its velocity with respect to a point C situated upon the perpendicular 

hr. 
to the line BP through B and at 100 ft. from B, when BP=$o ft. when 
BP= 1 50 ft. 

29. If a circular plate of metal expands by heat so that its diameter increases 
uniformly at the rate of T ^ of an inch per second, at what rate is its surface 
increasing when the diameter is 2 inches? 7r sq. in. 

100 sec. 

30. If the diameter of a sphere increaseSuniformly at the rate of y ¥ inches 

per second, what is its diameter when the volume is increasing at the rate of 5 cubic 

inches per second ? 10 

—y=r in. 
V 7t 

31. If the diameter D of the base of a cone increases uniformly at the 
rate of -^ inch per second, at what rate is its volume increasing when D becomes 
10 inches, the height being constantly one foot? cu. in. 

sec. 
32*. Water is poured into a conical glass, 3 inches in height, at a uniform 
rate, filling the glass in 8 seconds. At what rate is the surface rising at the end of 
1 second ? At what rate when the surface reaches the brim ? t in. , in. 

sec. sec. 

33. A train is running from A to B at the 
rate of 20 mi. an hour. The distance from A to C, 
on a perpendicular to A B, is two mi. Find the rate 
of the angle at C included between C A and a right 
line from C to the train. 

Let cp= variable angle at C. 
" y= distance from A to train. 

Given. CA=2mu, ^=20 — . 
dt hr. 

* Rice and Johnson's Calculus. 




EXAMPLES IN RATES. 89 



Required. 


d<p 

~dt- 
















cp .*. (£>=tan 


-1 JL 

CA 


and 


^/ cp 


CA 


2 


y=CA tan 


' AX: 2 +y 2 


~4+r J 




d(p 
~dl 


dcp 

-~dy 


dy 


2 


X 2 °= 


40 , 
4+7 3 






4-rj 2 




\ dt Jy=Q 


= 10. 

1 




V dt Jy: 


,2= 5 - 




V dt ) y-_ 


=0. 

=00 


The unit of 


measure 


of cp 


is a radian. 









34. In a right plane triangle one side adjacent to the right angle is 
constant and 4 mi. in length ; the other side adjacent to the right angle, denoted 
by y, is variable. Let cp represent the angle opposite y. Find the rate of cp ; 
first, when <p=f(y); also, when <p —I^X tan cp), corresponding to j=2mi M and 
explain the difference between the two results. 

Let ti=tang>= ^— — -_ <- • 
^ 4 



Rdy [dcp \ 1 du (dq> \ 



4. 
5 

cp changes less rapidly with respect to y than it does with respect to u, because y 
changes 4 times as fast as u. 

35. Determine the manner in which the sin of an angle varies with the angle. 

d€\xv (p dr 

— = cos <p=rate:=r, — — — sin <p. 

dcp dcp 

As cp increases from to - , the rate is +, but diminishing. Hence, the 
2 

sin increases, but its increments decrease. 

it 
From - to 7C, the rate is — , and diminishing. Hence, the sin diminishes and 
2 

its decrements increases numerically. 

"X it 
From 7t to - — , the rate is — , and increasing. That is, the sin decreases, 
2 \ 

but its decrements diminish numerically. 
o ft 
From - — to 2 tt, the rate is +, and increasing. That is, the sin increases, 
2 

and its increments increase. 

In a similar manner determine the circumstances of change of each trigono- 
metric function, with respect to the angle. 

36. Determine the rate of change of the tangent, regarded as a function of 
the sine of an angle. 



90 EXAMPLES IN RATES. 

Same of the sin as a function of the cos. 
Same of the sin as a function of the sec. 
Same of the cos as a function of the cot. 
Same of the cos as a function of the cosec. 
Same of the tan as a function of the sec. 
Same of the tan as a function of the versin. 
Same of the cosec. as a function of the covers. 
Same of the sec as a function of the vers. 

37*. Two points start together from an extremity of a diameter of a circle 
whose radius is 150 feet. One point moves uniformly along the diameter at the 
rate of 5 ft. per second ; the other moves in the circum. and is always in the 
perpendicular to the diameter through the first point. Find the velocity of the 
second point when the angle subtended by the arc described by it is 45 . 

38*. Two points start as in above example ; one moving uniformly along 
the tangent at the rate of 10 ft. per second and the other in the circum. so as to be 
always in the right line joining the first with the centre of the circle. Find the 
velocity of the second when passing the 45 point. 

* From Bowser's Calculus. 



CHAPTER VI. 

DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE 
VARIABLES. 

98. The Partial Differential of a Function of Two or more 
Variables, with respect to one of the variables, is the change that 
the function would undergo from any state, were it to retain its rate 
at that state, with respect to that variable, while that variable 
changed by its differential. 

The Total Differential of a Function of Two Variables is the 
change that the function would undergo from any state, were it to 
retain its rate at that state, with respect to each variable, while both 
variables changed by their differentials. 

Any function of two variables which changes uniformly with 
each variable, has a constant rate with respect to each, and its form 
must be some particular case of the general expression Ax-\-By-\-C, 

S 59- 

Representing such a function by z, we have 

z=Ax+By+C .... (i). 

Increasing x and y by their differentials, and denoting the 
corresponding new state of the function by z', we have 

z'=A(x+dx)+B(y+dy)+C .... (2). 
Subtracting (1) from (2), member from member, we have 

z'—z=Adx+Bdy. 
Since the function z changes uniformly with respect to each 
variable, the total differential of it, denoted by dz, is equal to the 
corresponding change in the function. 
Therefore, dz=Adx-\-Bdy. 

Adx is the corresponding partial differential of the function z 
with respect to x; and Bdy is the same with respect to y. 



92 DIFFERENTIATION. 

Hence, the total differential of any function of two variables, which 
changes uniformly with respect to each, is equal to the sum of the corre- 
sponding partial dijfere?itials . 

The total differential of any function of two variables which 
does not vary uniformly with each variable, is not, in general, the 
corresponding change in the function, but it is the corresponding 
change of a function having a constant rate with respect to each 
variable, equal to that of the given function at the state considered. 
In other words, the total differential is equal to that of a function 
which changes uniformly with each variable, and which has at the 
state considered its partial differentials equal to the corresponding 
partial differentials of the given function. 

Hence, the total differential of any function of two variables is 
equal to the sum of the corresponding partial differentials. 

That is, having z=f(x,y), then dz= -^ dx-\- -rdy. 

In a similar manner it may be shown that, the total differential of 
any function of any number of variables is equal to the sum of the 
corresponding partial differentials. 

Examples. 

1 . d (xy) = xdy +ydx. 

2. d{^ax 2 y — 2y 2 +3&x s — 5) = 6axydx-\-gl>x 2 dx+ ^ax 2 dy — 4ydy. 

( x+y \ _ 2 {xdy—ydx) 
3- d \ x _ y ) ~ {jc _ y) z 

4. d {x 2 y 2 z 2 )=2 y* z' d xdx+2x 2 z 2 ydy+2x 2 y 2 zdz. 

/ y\ _ xdy—ydx 

6. d [sin ixy)\—cos {xy) {ydx+xdy). 

ydx 

7 . d loge {xr) = — + log e xdy. 

( sin x dy \ 

8. dy s{n x =y sin * I \ Q g e y cos xdx+ ) * 

ydx — xdy 
y y<\/2 xy — x 2 
10. d sin (.r +j)=cos (x+y) (dx+dy). 



FUNCTIONS OF TWO VARIABLES. 



93 



II. 



2 =tan- 1 : 



y*+x'< 



Required — = 



dx \a' 1 — x' 2 

dv x 

12. u=y 2 +x-—a 2 =0. Required du=0, and^ = — -' 



dzt 



+ I2JT 2 — 24.X+24. Required ~r =e x x i 

14. tt— sin -1 {p — q), J>=3x, q=4x s . Required — = ., * 

dx /y/i — x s 



15. Deduce the formula ds= ydr 2 -\-r 2 dv' 1 , § 95, from the formulas, 

x=a+r cos v \ . — - 

y=b-\-rsmv \ L Anal - Geo -J; and ds—^/dx *+dy~, §90. 

16. One side of a rectangle increases at the rate of 3 in. per second and the 
other decreases at the rate of 2 in. per second. Find the rate of the area, when the 
first side is 10 in. and the second 8 in. in length. 

44 sq. in. per second. 
Z 



99. Let ATL be 

any surface, and 
AJ3CJD= u a portion 
of it included between 
the coordinate planes 
XZ, YZ, and the 
planes DQR BPS, 
parallel to them re- 
spectively. From § 25, 
we have u = f(x,y). 

Increase OP=x by 
PP'=h giving, § 53, 



d Ji = limit V f(x+h,y)—f{x^!) \ 
dx /im^>0 L h J 




Now increase OQ—y by QQ' '=&, giving, § 53, 

•**"* = limit [i 
dx dy km-^Q I 



limit r /(3+ft,y+*)-/fo y+k)-[f(x+h, y)-f(x, y)] ~| 



*R ,y 2 2/ {fit 

is a symbol for the partial differential coefficient of — , taken with 



dx dy 
respect to y. 



dx 



94 DIFFERENTIATION. 



i u limit r 

— — h^-^o • 
dv &s»-»o L 



-f(x+h, y+k)-f{x,y+k)-{f{x+h,y)-f{x,y)-Y 



dx dy fc^->0 

In which, 

f{x+h,y+k)-AEGI. f{x,y+k)=ABHI. 

f(x+/i,y)=A£FD. f(x,y)=ABGD. 

Hence, 

f(x+ h, y+k)—f(x,y+k)=AEGI—ABHI =BEGH. 

f (x+/i, y)—f{x, y)-AEFD—ABCD—BEFC. 

f(jc+/i,y+£)—f(x,y+k)—[f(x+/i,y)—f(x,y)]=BEG/f—BEEC = CEG/f. 

. d*u limit fCEGJ/1 

Therefore, -j—j- = h »»-»o — — — . 

dxdy jl'S^oL h k J 

Let CF' G'H' (not drawn) be the portion of the tangent plane 
to the surface at C included between the same planes that deter- 
mine C F GJF, and let j3 represent the angle between the tangent 
plane and XY. 

Then CF'G'JF'xcosj3=NJ?MS=/ik,§43,orCF'G'J?'=-^' 

From§ 44, we have k^o [ cpGH j =I . 

Hence § 31, Theorem X, 

d* u limit f CF' G'ff'l limit |~— ^1 1 

dx dy Jc s»-»o L h k J Jc ^->0 I j, % J cos p 

, d 2 u T 7 dxdy 
and -j— -dxdy— £■ 

ajf i/j/ T COS /J 

The same formula may be deduced in a similar manner from the 
figure without using the functional notation. 

Thus, -^ = 7 limit — -j — . Increasing y by QQ'=&, we have 



r Increment - / limit rJB CFIT]\-~\ 

d £ u _ H m i t due to A; ot U^<>L \ — S) 

dx dy k »»->0 I k ' ~ J 

limit T Increment ot *CFBrv\ ^ CFGH ^ 



T limit T Increment » 5 C.P^in .... „„ 

_ limit AS ^oL due to * of — ^J _ ^ V £F_ 

k-^^Y_ * J A»-»oL h 

limit rCF' G' H' 1 limit ["-ALTI z 

A»»-»0 L h k J A^-»0| ~XF"J cos P 



FUNCTIONS OF TWO VARIABLES. 



95 



100. Let ATL 

be any surface, and 
ABCJD—OJV=V J a 
volume limited by it, 
the three coordinate 
planes, and the planes 
DQR and BPS par- 
allel,respectively,to XZ 
and YZ. From § 28, 
we have V=/(x,y). 

By the method used 
in the last Article, con- 
sidering the corre- 
sponding volumes in- 
stead of the surfaces, 
we obtain. 

dxdy lc a»-»o L hk 

In which, 
f(xJ r /i,y+k)-/(x,y^k)-[/(x+Aj)-/(x,y)]=volCFGJI-JV.M. 




*■} 



Hence, § 45: 



.-= h?»->0 \- 
dx dy Jc m-X) L 



hk 



1 

-J =NC=: 



and 



- — -dx dy=z dxdy. 
dx dy y 



fi 



t°l 



D 174 







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